Question

solve for x. you may round your answer to 3 decimal places. 10^{x + 2} - 14 = 423

Explanation:

Step1: Isolate the exponential term

Add 14 to both sides of the equation \(10^{x + 2}-14 = 423\) to get \(10^{x + 2}=423 + 14\).
\(10^{x + 2}=437\)

Step2: Take the logarithm of both sides

Take the common logarithm (base 10) of both sides: \(\log(10^{x + 2})=\log(437)\).
Using the logarithm property \(\log(a^b)=b\log(a)\), we have \((x + 2)\log(10)=\log(437)\).
Since \(\log(10) = 1\), this simplifies to \(x + 2=\log(437)\).

Step3: Solve for x

Subtract 2 from both sides: \(x=\log(437)-2\).
Calculate \(\log(437)\approx2.6405\), so \(x\approx2.6405 - 2\).
\(x\approx0.6405\)

Step4: Round to 3 decimal places

Rounding \(0.6405\) to 3 decimal places gives \(x\approx0.641\).

Answer:

\(x\approx0.641\)