To solve the equation [tex]\( \log_5 (x^4) = 2.5 \)[/tex], let's follow these steps:
1. Use the properties of logarithms:
The given equation [tex]\( \log_5 (x^4) = 2.5 \)[/tex] can be rewritten using the properties of logarithms. One of the properties states that [tex]\( \log_b (a^c) = c \log_b (a) \)[/tex]. Applying this property:
[tex]\[
4 \log_5 (x) = 2.5
\][/tex]
2. Isolate [tex]\( \log_5 (x) \)[/tex]:
To find [tex]\( \log_5 (x) \)[/tex], divide both sides of the equation by 4:
[tex]\[
\log_5 (x) = \frac{2.5}{4}
\][/tex]
3. Simplify the fraction:
[tex]\[
\log_5 (x) = 0.625
\][/tex]
4. Rewrite the logarithmic equation in exponential form:
The equation [tex]\( \log_5 (x) = 0.625 \)[/tex] can be rewritten in its exponential form. By definition, [tex]\( \log_b (a) = c \)[/tex] means [tex]\( b^c = a \)[/tex]. Thus,
[tex]\[
x = 5^{0.625}
\][/tex]
5. Calculate [tex]\( 5^{0.625} \)[/tex]:
Using the calculated value,
[tex]\[
5^{0.625} \approx 2.734363528521053
\][/tex]
6. Round the result to the nearest hundredth:
Finally, rounding to the nearest hundredth,
[tex]\[
x \approx 2.73
\][/tex]
Therefore, the solution to the equation [tex]\( \log_5 (x^4) = 2.5 \)[/tex], rounded to the nearest hundredth, is:
[tex]\[
\boxed{2.73}
\][/tex]
