Solve the exponential equation e3 5x 16 Solve the logarith

Solve the exponential equation e^3 - 5x = 16 Solve the logarithmic equation log_9(x - 5) + log_9 (x + 3) = 1 Expand the logarithm using laws of laws log_5(3x^2/y^3) Combine the logarithm using law of logs 4 log x - 1/3 log(x^2 + 1) + 2 log (x - 1) Use change of base formula to evaluate the log. log_2^5

Solution

1. solution:

e^{(3 - 5x)} = 16

=> applying logarithm on both sides,

=> log e^{(3 - 5x)} = log 16

As per the rule, log e^(x) = x , so log e^{(3 - 5x)} = 3 - 5x

and => log 16 = log 2⁴   as per rule , log 2 ^x = x log2

        =>            = 4 log 2

         and as per change of base rule 4*log ²₂ = 4* log2 / log 2 = 4

=> log e^{(3 - 5x)} = log 16

=> 3 - 5x = 4

=> 3 = 5x + 4

=> 5x = 3 - 4

=> 5x = -1

=> x = -1/5

2) solution:

=> log ₉ ^{(x - 5)} + log ₉ ^{(x + 3)} = 1

=> As per rule of addition of logarithms for same base, log^a_x + log ^b _x = log ^(ab)_x

=> so left hand side will be converted based on addition of logarithms as the base is 9

=> log ₉^{[(x - 5)(x + 3)]} = 1

=> as per change of base rule log ^a _{b =} log a/ log b

=> log ₂^{[(x - 5)(x + 3)]} / log ₂⁽⁹⁾ = 1   [ after change of base rule, considering for all common base is 2)

=> log ₂^{[(x - 5)(x + 3)]} / log ₂^{(3 ^ 2)} = 1

=> log ₂^{[(x - 5)(x + 3)]} =   log ₂ ⁽⁹⁾

as per logarithmic equality rule ,   if log ₂ ^a = log ₂ ^b then   a = b.

=> so (x - 5)(x + 3) = 9

=>  x² - 2x - 15 = 9

=> x² - 2x - 15 - 9 = 0

=> x² - 2x - 24 = 0

=> x² - 6x - 4x - 24 = 0

=> x (x - 6) - 4 (x - 6) = 0

=> (x - 6) ( x - 4) = 0

=> (x - 6) = 0      and   (x - 4) = 0

=> x = 6   , x = 4

3). Solution:

=> log ₅^{(3 x^2 / y^3 )}

=> as per rule, log _x^(a/b) = log _x^a - log _x ^b

=> log ₅^{(3 x^2)} - log ₅ ^{(y^3)}

=> as per rule, log _x^(ab) = log _x^a + log _x ^b

=> log ₅³ + log ₅^{(x^2)} - log ₅ ^{(y^3)}

=> as per rule, log ₅ ^{(a^x)} = x log ₅ ^a

=> log ₅³ + 2 log ₅ ^x - 3 log ₅ ^(y)

=> as per rule,   log _b^a = log _x^a / log _x^b

=> (log 3 / log 5 ) + 2 (log x / log 5) - 3(log y / log 5)

=> (log 3 + 2 log x - 3 log y) / log 5

4). Solution:

=> 4 log x - (1/3) log (x² + 1) + 2 log (x - 1)

=> as per rule,   a log x = log x^a

=> log x⁴ - log (x² + 1) ^(1/3) + log (x - 1)²

=> as per logarithmic addition rule,     log a + log b = log (ab)

=> log x⁴ -   log [(x² + 1) ^(1/3) (x - 1)²]

=> as per logerithmic subtraction rule, log a - log b = log (a/b)

=> log    [x⁴ / ((x² + 1) ^(1/3) (x - 1)²) ]