To solve the equation [tex]\(3^x + 4^y = 5^z\)[/tex] for [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] and [tex]\(z\)[/tex], let's go through the steps methodically:
1. Understand the given equation:
[tex]\[
3^x + 4^y = 5^z
\][/tex]
Our goal is to express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex] and [tex]\(z\)[/tex].
2. Isolate the term involving [tex]\(x\)[/tex]:
We start by isolating [tex]\(3^x\)[/tex] on one side of the equation. To achieve this, we subtract [tex]\(4^y\)[/tex] from both sides:
[tex]\[
3^x = 5^z - 4^y
\][/tex]
3. Take the logarithm of both sides:
To solve for [tex]\(x\)[/tex], we take the logarithm of both sides. We'll use natural logarithms ([tex]\(\ln\)[/tex]) for simplicity:
[tex]\[
\ln(3^x) = \ln(5^z - 4^y)
\][/tex]
4. Utilize the properties of logarithms:
Recall that [tex]\(\ln(a^b) = b \ln(a)\)[/tex]. Using this property, we can simplify the left-hand side:
[tex]\[
x \ln(3) = \ln(5^z - 4^y)
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Finally, we solve for [tex]\(x\)[/tex] by dividing both sides by [tex]\(\ln(3)\)[/tex]:
[tex]\[
x = \frac{\ln(5^z - 4^y)}{\ln(3)}
\][/tex]
Therefore, the solution to the equation [tex]\(3^x + 4^y = 5^z\)[/tex] is:
[tex]\[
x = \frac{\ln(5^z - 4^y)}{\ln(3)}
\][/tex]
This formula allows you to determine the value of [tex]\(x\)[/tex] based on the given values of [tex]\(y\)[/tex] and [tex]\(z\)[/tex].
