Change of Base

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GRE Quantitative Reasoning › Change of Base

Questions 1 - 3

1

$Solve\ for\ x: 3^x=23$

Explanation

In order to solve a logarithm, we must first rewrite it in log form:

$3^x=23\ written\ in\ log\ form\ is: log_{3}(23)=x$

To solve for x, we must use the Change of Base:

$log_{3}(23)=\frac{log(23)}{log(3)}$

$\frac{log(23)}{log(3)}=2.85405$

This means that:

$3^{2.85405}=23$

2

Evaluate:

$3^{3x+3}=27^{2x+2}$

$\frac{1}{2}$

$-\frac{1}{2}$

$\frac{2}{3}$

$\frac{4}{13}$

Explanation

In order to evaluate the unknown variable, it is necessary to change the base. Looking at the right side of the equation, 27 is equivalent to three cubed.

Therefore, converting the right side of the equation to a base of 3 will allow setting both the left and right side of the exponential terms equal to each other.

$3^{3x+3}=27^{2x+2}$

$3^{3x+3}=3^{3(2x+2)}$

Log both sides to drop the exponents by log properties, and divide the log based 3 on both sides to cancel this term.

Solve for x.

3

Evaluate the following logarithm:

$log_{377}(563)$

Explanation

The simplest way to evaluate a logarithm that doesn't have base 10 is with change of base formula:

$log_ba=\frac{log_ca}{log_cb}$

So we have

$log_{377}(563)$

$log_{377}563=\frac{log_{10}563}{log_{10}377}\approx1.068$

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