Question

solve the following equations. if necessary, round your final answer to four decimal places.
a) $\log_{4}(5x + 9) = 3$
x =
b) $5^{x} = 22$
x =

Explanation:

Part (a)

Step1: Convert log to exponential form

Using the definition of logarithms: if $\log_b(a) = c$, then $b^c = a$. So for $\log_4(5x + 9) = 3$, we get $4^3 = 5x + 9$.

Step2: Calculate $4^3$ and solve for x

$4^3 = 64$, so the equation becomes $64 = 5x + 9$. Subtract 9 from both sides: $5x = 64 - 9 = 55$. Then divide by 5: $x = \frac{55}{5} = 11$.

Step1: Take log of both sides

Take the natural logarithm (ln) of both sides of $5^x = 22$: $\ln(5^x) = \ln(22)$.

Step2: Use logarithm power rule

Using the power rule $\ln(a^b) = b\ln(a)$, we get $x\ln(5) = \ln(22)$.

Step3: Solve for x

Divide both sides by $\ln(5)$: $x = \frac{\ln(22)}{\ln(5)}$. Calculate the values: $\ln(22) \approx 3.0910$, $\ln(5) \approx 1.6094$, so $x \approx \frac{3.0910}{1.6094} \approx 1.9206$.

Answer:

11

Part (b)