Solve for [tex][tex]$x$[/tex][/tex]. Round to the nearest hundredth when necessary.

[tex] \log _5 x^4 = 2.5 [/tex]

Answer: [tex] \square [/tex]



Answer :

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]