Question 8 Solve the equation $$ 4^{(x+5)}=7^{(x-2)} $$. Leave your answer in terms of logarithms (do not provide a decimal answer).

Question

UpStudy AI · Accepted Answer

We start with the equation

$$
4^{(x+5)} = 7^{(x-2)}.
$$

Taking the natural logarithm of both sides gives

$$
\ln\left(4^{(x+5)}\right) = \ln\left(7^{(x-2)}\right).
$$

Using the logarithm power rule, which states that $$\ln(a^b) = b\ln(a)$$, we can rewrite the equation as

$$
(x+5)\ln(4) = (x-2)\ln(7).
$$

Expanding both sides yields

$$
x\ln(4) + 5\ln(4) = x\ln(7) - 2\ln(7).
$$

To isolate the terms with $$x$$, subtract $$x\ln(7)$$ from both sides:

$$
x\ln(4) - x\ln(7) + 5\ln(4) = -2\ln(7).
$$

Next, subtract $$5\ln(4)$$ from both sides:

$$
x\ln(4) - x\ln(7) = -2\ln(7) - 5\ln(4).
$$

Factor out $$x$$ from the left-hand side:

$$
x\left(\ln(4) - \ln(7)\right) = -2\ln(7) - 5\ln(4).
$$

Finally, solve for $$x$$ by dividing both sides by $$\ln(4) - \ln(7)$$:

$$
x = \frac{-2\ln(7) - 5\ln(4)}{\ln(4) - \ln(7)}.
$$

This is the solution in terms of logarithms. Alternatively, by multiplying the numerator and denominator by $$-1$$, we can also write the answer as

$$
x = \frac{2\ln(7) + 5\ln(4)}{\ln(7) - \ln(4)}.
$$
$$ x = \frac{-2\ln(7) - 5\ln(4)}{\ln(4) - \ln(7)} $$

Question 8 Solve the equation \( 4^{(x+5)}=7^{(x-2)} \). Leave your answer in terms of logarithms (do not provide a decimal answer).