Question

which are possible first steps in solving the equation \\(4^{x + 3} = 18\\)?

rewrite each side with base 2.
subtract 18 from both sides.
take the base-4 logarithm of each side.
take the natural logarithm of each side.
take the common logarithm of each side.

Explanation:

Analyze the given equation

Using the Solving Exponential Equations Algebraically knowledge point

$$ 4^{x + 3} = 18 $$

Evaluate each proposed step

• Rewrite each side with base 2:

The left side can be written as \((2^2)^{x+3} = 2^{2x+6}\). However, \(18\) is not an integer power of \(2\) (\(18 = 2 \cdot 3^2\)), so we cannot easily rewrite both sides with a common base of \(2\) to solve it algebraically. Thus, this is not a practical first step.

• Subtract 18 from both sides:

This yields \(4^{x+3} - 18 = 0\). While mathematically valid, it does not help isolate the variable \(x\) from the exponent, so it is not a standard or helpful first step for solving.

• Take the base-4 logarithm of each side:

Applying \(\log_4\) to both sides:

$$ \log_4(4^{x+3}) = \log_4(18) \implies x + 3 = \log_4(18) $$

This successfully isolates the linear exponent, making it a valid and helpful first step.

• Take the natural logarithm of each side:

Applying \(\ln\) to both sides:

$$ \ln(4^{x+3}) = \ln(18) \implies (x+3)\ln(4) = \ln(18) $$

This allows us to solve for \(x\) using the power rule of logarithms, making it a valid and helpful first step.

• Take the common logarithm of each side:

Applying \(\log_{10}\) (or \(\log\)) to both sides:

$$ \log(4^{x+3}) = \log(18) \implies (x+3)\log(4) = \log(18) $$

This also allows us to solve for \(x\) using the power rule of logarithms, making it a valid and helpful first step.

Answer:

• Rewrite each side with base 2.
• Subtract 18 from both sides.
• Take the base-4 logarithm of each side. (Correct answer)
• Take the natural logarithm of each side. (Correct answer)
• Take the common logarithm of each side. (Correct answer)