To solve the equation [tex]\(5^{x-4}=7\)[/tex] for [tex]\(x\)[/tex] using the change of base formula, follow these steps:
1. Rewrite the equation in logarithmic form: To isolate [tex]\(x\)[/tex], we'll first take the logarithm of both sides of the equation. We can use the natural logarithm ([tex]\(\ln\)[/tex]) for this purpose.
[tex]\[
\ln(5^{x-4}) = \ln(7)
\][/tex]
2. Simplify the logarithmic expression: Use the logarithmic property [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex] to bring down the exponent.
[tex]\[
(x-4) \cdot \ln(5) = \ln(7)
\][/tex]
3. Isolate [tex]\(x-4\)[/tex]: To solve for [tex]\(x\)[/tex], divide both sides of the equation by [tex]\(\ln(5)\)[/tex].
[tex]\[
x-4 = \frac{\ln(7)}{\ln(5)}
\][/tex]
4. Calculate the logarithmic values: Let’s find the numerical values of [tex]\(\ln(7)\)[/tex] and [tex]\(\ln(5)\)[/tex].
- [tex]\(\ln(7) \approx 1.9459101490553132\)[/tex]
- [tex]\(\ln(5) \approx 1.6094379124341003\)[/tex]
5. Perform the division: Divide [tex]\(\ln(7)\)[/tex] by [tex]\(\ln(5)\)[/tex].
[tex]\[
\frac{1.9459101490553132}{1.6094379124341003} \approx 1.2090619551221675
\][/tex]
So,
[tex]\[
x-4 \approx 1.2090619551221675
\][/tex]
6. Solve for [tex]\(x\)[/tex]: Finally, add 4 to both sides of the equation to isolate [tex]\(x\)[/tex].
[tex]\[
x = 1.2090619551221675 + 4 \approx 5.209061955122167
\][/tex]
Hence, the solution to the equation [tex]\(5^{x-4}=7\)[/tex] is approximately [tex]\(x \approx 5.209\)[/tex].
