Sure, I'll explain the solution step-by-step for the given logarithmic equation [tex]\(\log_x 16 = \frac{4}{3}\)[/tex].
1. Understand the equation:
The equation [tex]\(\log_x 16 = \frac{4}{3}\)[/tex] tells us in logarithmic form that 16 is [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{4}{3}\)[/tex].
2. Rewrite in exponential form:
To make the problem easier to handle, convert the logarithmic form to its exponential form:
[tex]\[
x^{\frac{4}{3}} = 16
\][/tex]
3. Simplify the exponential equation:
Express 16 as a power of 2 because 16 can be written as:
[tex]\[
16 = 2^4
\][/tex]
Now the equation is:
[tex]\[
x^{\frac{4}{3}} = 2^4
\][/tex]
4. Isolate the base [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we need to get rid of the exponent [tex]\(\frac{4}{3}\)[/tex]. We do this by raising both sides of the equation to the reciprocal of [tex]\(\frac{4}{3}\)[/tex], which is [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
\left(x^{\frac{4}{3}}\right)^{\frac{3}{4}} = \left(2^4\right)^{\frac{3}{4}}
\][/tex]
5. Simplify the left side of the equation:
When you raise a power to a power, you multiply the exponents:
[tex]\[
x = 2^{4 \cdot \frac{3}{4}}
\][/tex]
6. Simplify the exponents:
In the exponent on the right side, [tex]\(4 \cdot \frac{3}{4} = 3\)[/tex]:
[tex]\[
x = 2^3
\][/tex]
7. Calculate the final value:
Evaluate [tex]\(2^3\)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\log_x 16 = \frac{4}{3}\)[/tex] is:
[tex]\[
x = 8
\][/tex]
