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Answer :
We start with the expression
[tex]$$
5^{\log_5 7} = x.
$$[/tex]
Notice that there is a logarithm in the exponent with the same base as the exponentiation. This allows us to use the fundamental identity for exponents and logarithms:
[tex]$$
a^{\log_a b} = b,
$$[/tex]
where [tex]$a > 0$[/tex], [tex]$a \neq 1$[/tex], and [tex]$b > 0$[/tex]. Here, [tex]$a = 5$[/tex] and [tex]$b = 7$[/tex]. Therefore, by applying the identity we immediately find
[tex]$$
x = 7.
$$[/tex]
For clarification, here are the step-by-step details:
1. Write down the given equation:
[tex]$$
5^{\log_5 7} = x.
$$[/tex]
2. Recognize the property that for any positive number [tex]$a$[/tex] (with [tex]$a \neq 1$[/tex]),
[tex]$$
a^{\log_a b} = b.
$$[/tex]
3. Substitute [tex]$a = 5$[/tex] and [tex]$b = 7$[/tex] into the property:
[tex]$$
5^{\log_5 7} = 7.
$$[/tex]
4. Thus, we conclude that
[tex]$$
x = 7.
$$[/tex]
This is the final answer.
[tex]$$
5^{\log_5 7} = x.
$$[/tex]
Notice that there is a logarithm in the exponent with the same base as the exponentiation. This allows us to use the fundamental identity for exponents and logarithms:
[tex]$$
a^{\log_a b} = b,
$$[/tex]
where [tex]$a > 0$[/tex], [tex]$a \neq 1$[/tex], and [tex]$b > 0$[/tex]. Here, [tex]$a = 5$[/tex] and [tex]$b = 7$[/tex]. Therefore, by applying the identity we immediately find
[tex]$$
x = 7.
$$[/tex]
For clarification, here are the step-by-step details:
1. Write down the given equation:
[tex]$$
5^{\log_5 7} = x.
$$[/tex]
2. Recognize the property that for any positive number [tex]$a$[/tex] (with [tex]$a \neq 1$[/tex]),
[tex]$$
a^{\log_a b} = b.
$$[/tex]
3. Substitute [tex]$a = 5$[/tex] and [tex]$b = 7$[/tex] into the property:
[tex]$$
5^{\log_5 7} = 7.
$$[/tex]
4. Thus, we conclude that
[tex]$$
x = 7.
$$[/tex]
This is the final answer.
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