Question

consider the following equation.
\log_{3} 4^{x}+3 = 4
find the value of x.
round your answer to the nearest thousandth.

Explanation:

Step1: Rewrite log in exponential form

Using $\log_b a = c \iff b^c = a$, we get:
$$4^{x+3} = 3^4$$
$$4^{x+3} = 81$$

Step2: Take log of both sides

Take $\log_4$ of both sides to isolate the exponent:
$$x+3 = \log_4 81$$
Use change of base formula $\log_b a = \frac{\ln a}{\ln b}$:
$$x+3 = \frac{\ln 81}{\ln 4}$$

Step3: Calculate the log value

Compute the numerical value:
$$\frac{\ln 81}{\ln 4} \approx \frac{4.39444915467}{1.38629436112} \approx 3.169925$$

Step4: Solve for x

Subtract 3 from both sides:
$$x \approx 3.169925 - 3$$

Answer:

$x \approx 0.170$