# Precalculus : Solve Logarithmic Equations

## Example Questions

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### Example Question #1 : Solve Logarithmic Equations

Evaluate a logarithm.

What is ?

Explanation:

The derifintion of logarithm is:

In this problem,

Therefore,

### Example Question #2 : Solve Logarithmic Equations

Using the inverse property to aid in solving.

Solve for ,

Explanation:

The natural logarithm and natural exponent are inverses of each other.  Taking the  of  will simply result in the argument of the exponent.

That is

Now, , so

### Example Question #3 : Solve Logarithmic Equations

Solve for  in the following logarithmic equation:

None of the other choices

None of the other choices

Explanation:

Using the rules of logarithms,

Hence,

So exponentiate both sides with a base 10:

The exponent and the logarithm cancel out, leaving:

This answer does not match any of the answer choices, therefore the answer is 'None of the other choices'.

### Example Question #4 : Solve Logarithmic Equations

Solve the following logarithmic equation:

Explanation:

In order to solve this equation, we must apply several properties of logarithms. First we notice the term on the left side of the equation, which we can rewrite using the following property:

Where a is the coefficient of the logarithm and b is some arbitrary base. Next we look at the right side of the equation, which we can rewrite using the following property for the addition of logarithms:

Using both of these properties, we can rewrite the logarithmic equation as follows:

We have the same value for the base of the logarithm on each side, so the equation then simplifies to the following:

Which we can then factor to solve for :

### Example Question #5 : Solve Logarithmic Equations

Solve the equation for .

Explanation:

We solve the equation as follows:

Exponentiate both sides.

Apply the power rule on the right hand side.

Multiply by .

Divide by .

### Example Question #1 : Solve Logarithmic Equations

Solve for :

Explanation:

First, simplify the logarithmic expressions on the left side of the equation:

can be re-written as .

Now we have:

.

The left can be consolidated into one log expression using the subtraction rule:

.

We now have log on both sides, so we can be confident that whatever is inside these functions is equal:

to continue solving, multiply by on both sides:

take the cube root:

### Example Question #7 : Solve Logarithmic Equations

.

Solve for

Explanation:

First bring the inside exponent in front of the natural log.

.

Next simplify the first term and bring all the terms on one side of the equation.

.

Next, let set

, so .

Now use the quadratic formula to solve for

and thus,  and .

Now substitute  with .

So,  since  and .

Thus,

### Example Question #2 : Solve Logarithmic Equations

Solve the logarithmic equation:

Explanation:

Exponentiate each side to cancel the natural log:

Square both sides:

Isolate x:

### Example Question #9 : Solve Logarithmic Equations

Solve for x:

Explanation:

The base of a logarithm is 10 by default:

convert to exponent to isolate x

subtract 1 from both sides

divide both sides by 2

Solve for x: