Question

An exponential function f(x)=(25)x
contains the point (3,0.064)
. Its corresponding logarithmic function g(x)=log25x
contains the point (a,3)
. What is the value of a?

Answer 1

To find the value of \( a \) for the logarithmic function \( g(x) = \log_{25}(x) \) that contains the point \( (a, 3) \), we use the relationship between exponential and logarithmic functions.

Given that \( f(x) = 25^x \) contains the point \( (3, 0.064) \), we can verify this point:

\[
f(3) = 25^3
\]

Calculating \( 25^3 \):

\[
25^3 = 25 \cdot 25 \cdot 25 = 625 \cdot 25 = 15625
\]

This means the point \( (3, 0.064) \) is incorrect, as \( 25^3 \neq 0.064 \).

Instead, let's check what \( 25^x \) would equal when \( x = 3 \):

\[
f(3) = 25^3 = 15625
\]

Now let's calculate what the logarithmic function \( g(x) = \log_{25}(x) \) looks like for the point \( (a, 3) \):

The point \( (a, 3) \) means:

\[
g(a) = 3
\]

By definition of the logarithm, this translates to:

\[
\log_{25}(a) = 3
\]

To solve for \( a \), we can rewrite this equation in exponential form:

\[
a = 25^3
\]

Now, we calculate \( 25^3 \):

\[
25^3 = 15625
\]

Thus, the value of \( a \) is:

\[
\boxed{15625}
\]