Logarithms and exponents - Algebra 2

is equivalent to . In this case, you know the value of (the argument of a logarithmic equation) and b (the answer to the logarithmic equation). You must find a solution for the base.

,

So,

This is a general formula that you should memorize. The inverse of is . You can use this formula to change an equation from a log function to an exponential function.

To solve this, you need to set up a logarithm. Our exponent is . The logarithm's base is . The value is the operand of the logarithm. Therefore, we can write an equation:

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

to...

You can put this into your calculator and get:

, or rounded,

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

to...

You can put this into your calculator and get:

, or rounded,

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

to...

You can put this into your calculator and get:

, or rounded,

To solve this, you need to set up a logarithm. Our exponent is . The number of which it is the exponent of is the base. This is the logarithm's base. The value is the operand of the logarithm. Therefore, we can write an equation:

Now, you cannot do this on your calculator. Therefore, using the rule for converting logarithms, you need to change:

to...

You can put this into your calculator and get:

, or rounded,

For logarithmic equations, can be rewritten as .

In this expression, is the base of the equation (). is the exponent () and is the term ().

In putting each term in its appropriate spot, the exponential equation can be converted to .

Solve the following logarithm:

Recall that we can convert logarithms to exponential form via the following:

Using this approach, convert the given log to exponential form:

Rewrite the following expression as an exponential expression:

Recall the following property of logs and exponents:

Can be rewritten in the following form:

So, taking the log we are given;

We can rewrite it in the form:

So b must be a really huge number!

Convert the following logarithmic equation to an exponential equation:

Recall the following:

This

Can be rewritten as

So, our given logarithm

Can be rewritten as

Fortunately we don't need to expand, because this woud be a very large number!

Rewrite the follwing equation as a logarithm:

To complete this problem, recall the following relationship:

can be rewritten as

So, this:

Is the same thing as this:

When the base is raised to a certain power, taking the natural log of this whole term will eliminate the exponential and the power can be pulled out as the coefficient.

The answer is:

In order to solve this log, we will need to write 125 in terms of one fifth to a certain power.

Rewrite 125 as an exponent of one-fifth.

According to the log rule, , the bases will cancel, leaving just the exponent.

The answer is:

Notice that the term in the log can be rewritten as a base raised to a certain power.

Rewrite the number in terms of base five.

According to log rules, the exponent can be dropped as a coefficient in front of the log.

The answer is:

Evaluate the log using the following property:

The log based and the base of the term will simplify.

The expression becomes:

The answer is:

By definition of a logarithm,

if and only if

Take the th root of both sides, or, equivalently, raise both sides to the power of , and apply the Power of a Power Property:

or

By definition, it follows that , so the statement is true.

By definition,

If and only if

Square both sides, and apply the Power of a Power Property to the left expression:

It follows that for all positive not equal to 1,

for all .

By definition, if and only if . However,

,

making this false.

By definition, if and only if .

8 and 16 are both powers of 2; specifically, . The latter equation can be rewritten as

By the Power of a Power Property, the equation becomes

or

It follows that

,

and

,

the correct response.