# Precalculus : Properties of Logarithms

## Example Questions

### Example Question #21 : Properties Of Logarithms

Solve the equation:

Explanation:

We notice that all of the bases are powers of 2 so we can re-write them as 2 raised to a certain power.

Then we simplify both sides of the equation so we have a single base 2 raised to a power. When we have when we have exponentials with the same base multiplied we add the exponents, when we have them raised to an exponent we multiply the exponents.

Now that we have the same base on both sides we set the exponents equal and solve for x.

### Example Question #22 : Properties Of Logarithms

The logarithms for this question are common logs (i.e. base 10).

Solve for  in the following expression.  Round to three decimal places.

Explanation:

The expression can be simplified using logarithmic properties.

We use the definition of common logarithm to solve for x.

So x = 21.544 -5 or 16.544.

### Example Question #23 : Properties Of Logarithms

Solve for the value of x by simplifying the following logarithmic function:

Explanation:

Log Property: Log(x) + Log(y) = Log(xy)

The equation  can be simplified using the log property stated above.

The simplified version of the equation is:

Substitute Log(10) in for 1, since Log(10) = 1

Next, remove the logs from both sides

Solve for the value of x:

A negative number cannot be evaluated in the log function equation above so the answer is

### Example Question #24 : Properties Of Logarithms

Find the value of x to satisfy .

Explanation:

First, remove the log from the equation by changing the form:

Next, change both sides so that they share a common base. In this case the common base is 4.

Reduce the amount of exponents on the left side. Since an exponent is raised to an exponent you multiply the exponents together.

Set the exponents on both sides equal to eachother and solve for x.

### Example Question #25 : Properties Of Logarithms

Simplify

.

Explanation:

Since the logarithmic function  is the inverse of the exponential function , we can make a straightforward simplification as follows:

### Example Question #26 : Properties Of Logarithms

Solve the equation,

.

Explanation:

Using the logarithmic product rule , we simplify as follows:

### Example Question #21 : Properties Of Logarithms

Simplify the expression .

Explanation:

Utilizing the power rule of logs we reduce the original expression to:

From there we use the product property to obtain:

Since 4 to the second power is 16 this expression yields

, then we multiply through by the one half to obtain our final answer .

### Example Question #28 : Properties Of Logarithms

is the simplified form of which of the following expressions?

Explanation:

First use the power property of logs to get:

.

Then use the quotient property to get:

.

Then finally use the product property and simplify to yield the final answer, .

### Example Question #29 : Properties Of Logarithms

Simplify.

Explanation:

We observe that  and  and substitute to yeild

Now we use the fact that  to simplify

Since  simplifying again yeilds

which is just

This yeilds a final answer of 36

### Example Question #30 : Properties Of Logarithms

Simplify the following expression so that it is only 1 logarithm: