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( Inquiry Question: )

Notes:

Recall: What is the inverse of an Exponential Function?

Solving a Logarithmic Equation

Let’s solve the equation log (x + 3) = log (3x +1).

First we set the inside of the logs equal to each other, which will look like:

From here, we simply solve for x by combining our like terms.

And then we have our solution.

Let’s try another! And another!

ln (4 – x) = ln (4x – 11) (x – 1) = (2x + 1)

Solving by Equating the Exponents

Just like solving the logarithmic equation, solving logs with exponents of the same base is done by setting the arguments equal to each other.

Let’s solve

First, we rewrite each power so that they have a base in common.
Next, we simplify our exponents by applying the power of a power property.
Now that we have the same base, we set the exponents equal to each other.
Last but NOT least, we solve for x

A few more examples.

You try.

Solving by Rewriting with Logarithms

One way we can solve exponential equations is by taking the exponential equation and rewriting it in logarithm form. Remember:

How about we go through an example:

Solve

( Step 1: Get the base by itself. With this example, we divide both side of the equation by 2. Step 2: Rewrite the equation in logarithmic form. Step 3: Use the change of base formula to evaluate the log. Plug this into the calculator. Step 4: Solve for x. )

One more.

Solve

( Step 1: Get the base by itself. With this example, we divide both side of the equation by 10. Step 2: Rewrite the equation in logarithmic form. Step 3: Remember that is natural log, or ln . Replace it. Step 4: Solve for x. )

Solve each equation for x.

1. 2. 3.

Solving by Rewriting with Exponents

Last but not least, we can use logs to solve by setting them as a base to each side of an exponential equation.

To solve we need to do the following:

( Step 1 - Get the log by itself; divide each side by the number in front of the log. Step 2 - Rewrite each side in exponential form. Step 3 - Now, we solve for x. )

Now try some on your own!

1. 2.

3. 4 ln 2x = 5 4.