1954 AMC 12 Problem 38

Attempt Problem 38 of the 1954 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1954 AMC 12 solutions, or check the answer key.

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38.

If

\log 2=0.3010

and

\log 3=0.4771,

the value of

x

when

3^{x+3}=135

is approximately:

5

1.47

1.67

1.78

1.63

Answer: B

Concepts:logarithm exponent estimation

Difficulty rating: 1340

Small hint:

Divide by

3^3

to obtain

3^x=5

Big hint:

Use

\log5=1-\log2

and divide by

\log3

Solution:

The equation reduces to

3^x=5,

so

\begin{aligned} x&=\frac{\log5}{\log3}\\ &=\frac{1-\log2}{\log3}\\ &=\frac{0.6990}{0.4771}\\ &\approx1.47. \end{aligned}

Thus, the correct answer is B.

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