Question
Write the equation in its equivalent exponential form. Then solve for \( x \). \( \log _{4}(x-5)=3 \) \( \log _{4}(x-5)=3 \) in its equivalent exponential form is \( \square \). (Do not simplify.)
Ask by Ayala Pena. in Saudi Arabia
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equivalent exponential form is \( 4^3 = x - 5 \), and the solution for \( x \) is \( 69 \).
Solution
To convert the logarithmic equation \( \log_{4}(x-5) = 3 \) into its equivalent exponential form, we follow the definition of logarithms. The equation states that \( 4 \) raised to the power of \( 3 \) equals \( x - 5 \).
Thus, the equivalent exponential form is:
\[
4^3 = x - 5
\]
Now, we can solve for \( x \):
1. Calculate \( 4^3 \):
\[
4^3 = 64
\]
2. Set the equation:
\[
64 = x - 5
\]
3. Add \( 5 \) to both sides to isolate \( x \):
\[
x = 64 + 5
\]
4. Therefore:
\[
x = 69
\]
So, the equivalent exponential form is \( 4^3 = x - 5 \) and the solution for \( x \) is \( 69 \).
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Simplify this solution The Deep Dive
In its equivalent exponential form, the equation \( \log _{4}(x-5)=3 \) translates to \( x - 5 = 4^3 \). Now, calculating the right side gives \( x - 5 = 64 \). Adding 5 to both sides, we find \( x = 64 + 5 = 69 \). Thus, the value of \( x \) is \( \boxed{69} \).
