Exercise 23 Page 461

To evaluate the given logarithm, we will start by writing a logarithmic equation. log_()15 625=x In order to solve this equation, we can rewrite it as an exponential equation by using the definition of a logarithm. log_b x=y ⇔ x= b^y The above means that the logarithm y is the exponent to which b must be raised to get x. For our exercise, y is the exponent to which 15 must be raised to get 625. log_(15) 625=x ⇔ 625=( 1/5)^x Finally, to solve the exponential equation, we will rewrite the terms so that they have a common base.

625 =(1/5)^x

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Write as a power

WritePow

Write as a power

5^4= (1/5)^x

SplitIntoFactors

Split into factors

5^(-1(-4)) = (1/5)^x

ProdInExponent

a^(m* n)=(a^m)^n

(5^(-1))^(-4)=(1/5)^x

NegOneExponentToFrac

a^(- 1)=1/a

(1/5)^(- 4)=(1/5)^x

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. ( 15)^(- 4)=( 15)^x ⇔ x=- 4