SOLUTION: prove that : {{{25^(log8)*8^(log4)}}}{{{""=""}}}{{{64}}}

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: prove that : {{{25^(log8)*8^(log4)}}}{{{""=""}}}{{{64}}}       Log On


   

Question 1096402: prove that : 25%5E%28log8%29%2A8%5E%28log4%29%22%22=%22%2264
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Prove:
25%5E%28log%28%288%29%29%29%2A8%5E%28log%28%284%29%29%29%22%22=%22%2264

Start with the left side and show that it equals 64

25%5E%28log%28%288%29%29%29%2A8%5E%28log%28%284%29%29%29%22%22=%22%22

Write 5 as 5², 8 as 2³, and 4 as 2²

%285%5E2%29%5E%28log%28%282%5E3%29%29%29%2A%282%5E3%29%5E%28log%28%282%5E2%29%29%29%22%22=%22%22

Use the rule of logarithms that allows an exponent of what
a log is taken of to be moved out front as a coefficient of
the logarithm. 

%285%5E2%29%5E%283log%28%282%29%29%29%2A%282%5E3%29%5E%282log%28%282%29%29%29%22%22=%22%22

Use the rule of exponents that says that when a power
is raised to a power, we multiply the exponents

5%5E%286log%28%282%29%29%29%2A2%5E%286log%28%282%29%29%29%29%22%22=%22%22

Use the rule that says if the product of two powers
have the same exponent, we can multiply the bases
and use the common exponent. 

%285%2A2%29%5E%286log%28%282%29%29%29%22%22=%22%22

use 5×2 = 10

10%5E%286log%28%282%29%29%29%22%22=%22%22

Use the rule that says that a coefficient of a logarithm
can be moved to the exponent of what the logarithm is taken of:

matrix%282%2C1%2C%22%22%2C10%5E%28log%28%282%5E6%29%29%29%29%22%22=%22%22

Use the definition of "log(26)", which is "the power to which
10 must be raised to give 26".  Since 10 here is
raised to that power, its value must be 26.

2%5E6%22%22=%22%22

64

proved

Edwin