Question

An exponential function f(x)=(25)x
𝑓
(
𝑥
)
=
(
2
5
)
𝑥
contains the point (3,0.064)
(
3
,
0.064
)
. Its corresponding logarithmic function g(x)=log25x
𝑔
(
𝑥
)
=
log
2
5
⁡
𝑥
contains the point (a,3)
(
𝑎
,
3
)
. What is the value of a?

Answer 1

To find the value of \( a \) where the logarithmic function \( g(x) = \log_{25}(x) \) contains the point \( (a, 3) \), we start by interpreting this point. The point \( (a, 3) \) means that:

\[
g(a) = 3
\]

This can be translated into the logarithmic equation:

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

Using the properties of logarithms, we can rewrite this in exponential form:

\[
a = 25^3
\]

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

\[
25 = 5^2 \implies 25^3 = (5^2)^3 = 5^{2 \times 3} = 5^6
\]

Now, let's compute \( 5^6 \):

\[
5^6 = 15625
\]

Thus, the value of \( a \) is

\[
\boxed{15625}
\]