LOGARITHM

(1). Evaluate log₁₀ 25 +  log₁₀ 32 –  log₁₀ 8

(A) 0.2 (B) 2 (C) 100 (D) 409

Solution:-Using CASIO fx-991MS calculator

[log] [2] [5] [+] [log] [3] [2] [-] [log] [8] [=] (2)

The correct option is “B”

(2). Evaluate log₅ 62.5 – log₅ ½

(A) 3, (B) 4, (C) 5, (D) 125

Solution:-Using CASIO fx-991MS calculator

But log_a b = log₁₀ b

.                  log₁₀ a

--when dividing the numerator and the denominator put brackets to avoid error i.e.

(log 62.5)            (log ½)

.(log 5)                (log 5)

[ON] [(] [log] [6] [2] [.] [5] [)] [a^b/_c] [(] [log] [5] [)] [-] [(] [log] [1] [a^b/_c] [2] [)] [=] (3)

The correct option is “A”

(3). Solve log₃ (4x + 1) – log₃ (3x – 5) = 2

(A) 2 (B) 4 (C) -2 (D) 3

Since:-Using CASIO fx-991MS calculator cannot solve base 3 apply

log_a b = log₁₀ b 

              log₁₀ a

i.e. log (4x + 1)            log₁₀ (3x – 5)

            log 3                      log₁₀ 3

left axis is = right axis, left axis =  log₃ (4x + 1) - log₃ (3x – 5), right axis = 2

if option A,B,C,D gives you 3 it is correct but if it doesn't give 3 it is wrong

checking option (A) 2    put x=4 in the question given

[ON] [2] [SHIFT] [STO] [X] (x=2)

Note:-Please use the key [AC] during calculations after you press the key [ON] on any question given                          
--when dividing the numerator and the denominator put brackets to avoid error when using CASIO fx-991MS calculator 

i.e. “(log (4x + 1)        (log₁₀ (3x – 5))”

           (log 3)                 (log₁₀ 3)

[AC] [(] [log] [(] [4] [ALPHA] [X] [+] [1] [)] [)] [a^b/_c] [(] [log] [3] [)] [-] [(] [log] [(] [3] [ALPHA] [X] [-] [5] [)] [)] [/] [(] [log] [3] [)] [=] (2)

Option “A” is correct. To prove that option B, C, D is wrong
Checking option (B) 4   put x=4 in the question given
[AC] [4] [SHIFT] [STO] [X](x=4)
Press [↑] [=] (0.807658174)
Option “B” is wrong
Checking option (C) -2    put x= -2 in the question given
[AC]  [(-)] [2] [SHIFT] [STO] [X](X= -2)
Press [↑] [=] (Math error)
Option C is wrong
Checking option (D) 3  put x=3 in the question given
[AC] [3] [SHIFT] [STO] [X](X=3)
Press [↑] [↑] [=] (1.072858012)
Option D is wrong

(4). solve for x in the equation log₂ (¹/₁₆)^{2x + 3} = log₂ 8^1/3(5x+1)
(A) -2, (B) 1, (C) -1, (D) 2
Solution:-Using CASIO fx-991MS calculator
when raising a fraction equation to the power put brackets to avoid error
--when dividing the numerator and the denominator put brackets to avoid error when using CASIO fx-991MS calculator
But log_a b = log₁₀ b/log₁₀ a.  Applying it in the question
log (¹/₁₆)^{2x + 3}  =  log  8^1/3(5x+1)
log 2                     log 2
Left axis = log₁₀ (¹/₁₆)^{2x + 3} ,         right axis = log₁₀ 8^1/3(5x+1)
                                log₁₀ 2                                      log₁₀ 2
When dividing the numerator and the denominator put brackets to avoid error when using CASIO fx-991MS calculator
i.e. “(log (¹/₁₆)^{2x + 3}) = (log₂ 8^1/3(5x+1))”
            (log 2)                    (log 2)
To trace the option make x = any of the option given i.e. x = any of the option A, B, C, D  if the left axis is equal to the right axis it is correct but if the left axis is not to right axis it is wrong
Checking option (A) 2    make x = 2
[ON] [2] [SHIFT] [STO] [X] (x=2)
Note:-Please use the key [AC] during calculations after you press the key [ON] on any question given                                                                                                                        
[AC] [(] [log] [(] [1] [a^b/_c] [1] [6] [)] [^] [(] [2] [ALPHA] [X] [+] [3] [)]  [)] [a^b/_c] [(] [log] [2] [)] [=] (-28) [(] [log] [8] [^] [(] [1] [a^b/_c] [3] [(] [5] [ALPHA] [X] [+] [1] [)] [)] [)] [a^b/_c] [(] [log] [2] [)] [=] (11)
Left axis is not equal to right axis option (A) is wrong
Checking option (B) 1    make x = 1
[AC] [1] [SHIFT] [STO] [X] (x=1)
Press [↑] [↑] [=] (-20)
Press [↑] [↑] [=] (6) 
Left axis is not equal to right axis option (B) is wrong
Checking option (C) -1    make x = -1
[AC] [(-)] [1] [SHIFT] [STO] [X] (x=-1)
Press [↑] [↑] [=] (-4)
Press [↑] [↑] [=] (-4) 
Left axis is equal to right axis option (C) is correct
Checking option (D) 2    make x = 2
[ON] [2] [SHIFT] [STO] [X] (x=2)
Press [↑] [↑] [=] (4)
Press [↑] [↑] [=] (-9) 
Left axis is not equal to right axis option (D) is wrong   
The correct option is “C”

(5). Given that log₁₀ 7 = x and log₁₀ 2 = y evaluate log₁₀ 35
(A) x + y (B) x – y + 1 (C) ^{x + 1}/_y (D) x – y
Solution:-Using CASIO fx-991MS calculator
log₁₀ 7 = x and log₁₀ 2 = y
[ON] [log] [7] [SHIFT] [STO] [X] (X=0.84509804)
Note:-Please use the key [AC] during calculations after you press the key [ON] on any question given                            
[AC] [log] [7] [SHIFT] [STO] [X]  (X=0.84509804)
[AC] [log] [2] [SHIFT] [STO] [Y] (Y=0.301029995)
[AC] [log] [3] [5] [=] (1.544068044)
if option A,B,C,D gives you 1.544068044 it is correct but if it doesn't give 1.544068044 it is wrong
Checking option (A) x+ y
[AC] [ALPHA] [X] [+] [ALPHA] [Y] [=] (1.146128036)
Option (A) is wrong because it did not give 1.544068014
Checking option (B) x – y + 1
[AC] [ALPHA] [X] [-] [ALPHA] [Y] [+] [1] [=] (1.146128036)
Option (B) is correct because it give 1.544068014
Checking option (C) ^{x +1}/_y
[AC] [(] [ALPHA] [X] [+] [1] [)] [a^b/_c] [ALPHA] [Y] [=] (6.129283017)
Option (C) is wrong because it did not give 1.544068014
Checking option (D) x – y
[AC] [ALPHA] [X] [-] [ALPHA] [Y] [=] (0.544068044)
Option (D) is wrong because it did not give 1.544068014
The correct option is “B”

(6). Given that log₄ (y – 1) + log₄ ¹/_2x = - ½ and log₄ (y + 1) + log₂ x = 2 solve for x and y respectively
(A) 2, 4 (B) 3, 2 (C) -2, -3 (D) 2, 3
Solution:-Using CASIO fx-991MS calculator
Make x = any of the option given i.e. x = any of option (A), (B), (C), (D)
log₄ (y – 1) + log₄ ¹/_2x =   - ½ …………………………………………(eq 1)
log₄ (y + 1) + log₂ x =    2 ……………………………………………...(eq 2)
left axis = log₄ (y – 1) + log₄ ¹/_2x =   - ½ …………………………………………(eq 1)
left axis = log₄ (y + 1) + log₂ x =    2 ……………………………………………...(eq 2)
Right axis = - ½.......................................................................................................(eq 1)
Right axis = 2...........................................................................................................(eq 2)
left axis must be equal to right axis if left axis is not equal to right axis the option is wrong                              
but log_a b = log₁₀ b/log₁₀ a
applying it in equation 1 and equation 2
log₁₀ (y – 1)  +  log₁₀ ¹/_2x  =   - ½ …………………………………………(eq 1)
  log₁₀ 4              log₁₀ 4
log₁₀ (y-1) + log₁₀ ¹/_2x  = 2 ……………………………………………..(eq 2)
 log₁₀ 4           log₁₀ 2
when dividing the numerator and the denominator put brackets to avoid error when using CASIO fx-991MS calculator
i.e. “(log₁₀ (y-1)) + (log₁₀ ¹/_2x)”
         (log₁₀ 4)          (log₁₀ 4)
--when dividing the numerator and the denominator involving multiples put brackets to avoid error i.e. ¹/_(2x)  except it is a whole numbers when using CASIO fx-991MS calculator
Checking option (A) 2, 4  make x = 2 y = 4
[ON] [2] [SHIFT] [STO] [X] (x=2)
Note:-Please use the key [AC] during calculations after you press the key [ON] on any question given                                                                                        
[AC]  [4] [SHIFT] [STO] [Y] (y=4)
[AC] [(] [log] [(] [ALPHA] [Y] [+] [1] [)] [)] [a^b/_c] [(] [log] [4] [)] [+] [(] [log] [1] [a^b/_c] [(] [2] [ALPHA] [X] [)] [)] [a^b/_c] [(] [log] [4] [)] [=] (-2.07518749)
Option (A) is wrong because equation 1 did not give us – ½ don’t bother to go to equation 2 go to option (B)                                                                                                  
Checking option (B) 3, 2  make x = 3 y = 2
[AC] [3] [SHIFT] [STO] [X] (x=3)
[AC]  [2] [SHIFT] [STO] [Y] (y=2)
Press [↑] [↑] [=] (-1.29248125)
Option (B) is wrong because equation 1 did not give us – ½ don’t bother to go to equation 2 go to option (C)             
Checking option (C) -2, -3  make x = -2 y = -3
[ON] [(-)] [2] [SHIFT] [STO] [X] (x=-2)
[ON] [(-)] [3] [SHIFT] [STO] [Y] (y=-3)
Press [↑] [↑] [=] (math error)
Option (C) is wrong because equation 1 did not give us – ½ don’t bother to go to equation 2 go to option (D)
Checking option (D)2, 3  make x = 2 y = 3
[AC] [2] [SHIFT] [STO] [X] (x=2)
[AC]  [3] [SHIFT] [STO] [Y] (y=3)
Press [↑] [↑] [↑] [↑] [=]  (-0.5)
Since equation (1) gives – ½  Impute equation (2)
[AC] [(] [log] [ALPHA] [(] [Y] [+] [1] [)] [)] [a^b/_c] [(] [log] [4] [)] [+] [(] [log] [ALPHA] [X] [)] [a^b/_c] [(] [log] [2] [)] [=] (2)
Left axis is = right axis the correct option is “D”

Unforgiveness is a sin, if you don’t forgive; your trespasses(sins) will not be forgiven

Reason. In Mathew 6:14-15- if ye forgive men their trespasses your heavenly father will also forgive your trespasses but if ye forgive not men their trespasses neither will your father forgive your trespasses

Also in Mark 11:25-26- And when ye stand praying forgive if you have ought against any that your father also which is in heaven will forgive your trespasses. But if ye do not forgive, neither will your father which is in heaven forgive your trespasses.

For your heavenly father to forgive your trespasses. You must forgive those that trespass against you including your enemies. In Matthew 5:44-Love your enemies, bless them that curse you, do good to them that hate you, pray for them which despitefully use you and persecute you. As a Christian/Believer of JESUS CHRIST you must forgive your enemies including those that commit evil against you because if you don’t forgive them you are in darkness. In 1 John 2:9-11 – He that saith he is in the light and hateth his brother is in darkness even until now. He that loveth his brother abideth in the light and there is none occasion of stumbling in him. But he that hateth his brother is in darkness and walketh in darkness and knoweth not wither because that darkness has blinded his eyes. A believer will say I am in the light by being a deacon, choir, usher, pastor, evangelist, But know that Christians/believers who is indulge in unforgiveness will not inherit the kingdom of God and they will spend their eternity in hell (the fire that never shall be quenched). Mark 9:43

Know that if you don’t forgive today, your prayers will not be answered. But if you are touched by the word of GOD on unforgiveness and you have make a decision in your heart to forgive whomsoever wronged you, offend you, please make this confession

LORD JESUS, as I come to you, have mercy upon me, as I forgive those that offend me today. Forgive me my trespasses; receive me into your kingdom, IN JESUS NAME AMEN.

NOW: that you have confess this prayer of unforgiveness you are free to confess the next prayer below if you are not

Born-again

if you have not born-again or given your life to Jesus Christ say this prayer “Lord JESUS, as I come to you today, Have mercy upon me, Forgive me all my sins, wash me with your precious blood, save my soul, as I receive you today as my lord and personal savior, write my name in the book of life, IN JESUS NAME AMEN.

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