Evaluating Logarithms

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GRE Quantitative Reasoning › Evaluating Logarithms

Questions 1 - 5

1

Solve:

Explanation

Step 1: Re-write the log equation as an exponential equation. To do this, take the base of the log function and raise it to the number on the right side of the equal sign. This new exponent is equal to the number to the right of the log base.

Step 2: Re-write the right hand side as a power of 4..

Step 3: Re-write the equation

Step 4: We have the same base, so we can equal the exponents..

2

Solve for :

Explanation

Use rules of logarithms...

Take the base of the log and raise it to the number on the right side of the equal sign (which becomes the exponent):

$\4^x=16^3 \4^x=(4^2)^3 \4^x=4^6 \x=6$

3

$Solve\ for\ x: log_{x}(49)=2$

$\pm7$

Explanation

In order to solve for x, we must first rewrite the log in exponential form.

Every log is written in the below general form:

$log_{base}(result)=exponent\ is\ rewritten\ as: base^e^x^p^o^n^e^n^t=result$

In this case we have:

$log_{x}(49)=2$

This becomes:

We can solve this by taking the square root of both sides:

$\sqrt{x^2}=\sqrt{49}$

$x=\pm7$

4

Evaluate: $\log_6 46656=x$

Explanation

Step 1: Write the expression in exponential form...

Given: $\log_a b=x\implies a^x=b$

$\log_6 46656=x\implies 6^x=46656$

Step 2: Convert the right hand side into a power of 6..

Step 3: Re-write the equations...

Since the bases are equal, taking log of both sides will cancel them.

$\log_6 6^x=\log_6 6^6$

$\implies x=6$

5

$Solve\ for\ x:\ log_{2}(8)=x$

Explanation

$In\ order\ to\ solve\ for\ x\ we\ must\ know\ how\ to\ rewrite\ a\ log\ in\ exponent\ form:$

$We\ can\ use\ the\ general\ rule:\ log_{b}(a)=x\ when\ rewritten\ in\ exponential\ form\ we\ get: b^{x}=a$

$To\ Solve\ for\ x:\ log_{2}(8)=x;\ we\ rewrite\ as\ 2^{x}=8$

$We\ know\ that\ 2^{3}=8;\ so\ the\ value\ of\ x=3$

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