Logarithmic Form: Converting & Overview

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

Logarithmic form is useful when you want to solve for a variable in an exponent. In this lesson, you will learn how to change from exponential to logarithmic form.

Exponential Equations

Sometimes you may have an exponential equation - an equation with a variable in an exponent - and you want to solve for that variable. For example, suppose you have any of the following equations:

Several exponential equations

These three equations have in common that x is in the exponent. They are all exponential equations. In all three cases, solving for x is tricky - in fact, it is only possible if you use logarithms.

Logarithms and Inverses

You might remember that solving for (or 'isolating') a variable in an algebraic equation means using an inverse. For example, if you have y = 3x, the x is multiplied by 3. To isolate the x, you have to do the opposite of multiplying by 3. You have to divide by 3. Multiplication and division are inverse, or opposite, procedures. One undoes the effect of the other. Similarly, addition and subtraction are inverse operations. If you have y = x + 5, then you subtract 5 (opposite of adding 5), to isolate the x.

It's a bit like following a complex set of directions you printed off to get from your house to pick up the date you met on the Internet. Getting there is not too bad. But, when you meet your date and realize he is about 30 years older than the person you thought you were meeting, getting back home is a little more complicated. You have to follow the directions backward. You have to do the opposite, or inverse, of everything you did to get there.

If you are raising a number to a power (for example, 3^x), then the opposite of raising the number to the power is taking the logarithm of it. Just like you solve y = 7x by dividing both sides by 7, you solve y = 10^x by taking the logarithm (base 10) of both sides.

Another approach to solving for a variable in an exponent is to convert the exponential equation to logarithmic form - rewrite it as a logarithmic equation.

Logarithmic Equations

A logarithmic equation is simply an equation with a logarithm in it - and a variable inside the log part.

For example, these are all logarithmic equations:

Several logarithmic equations

Exponential and Logarithmic Form

Every equation that is in exponential form has an equivalent logarithmic form, and vice versa.

For example, the following two equations are equivalent:

Exponential and log form

Both equations have a 'b,' the base, an x, and a y.

These two equations are equivalent, just like these two equations are equivalent: y = x + 9 and y - 9 = x. Using algebra, you can get from one to the other.

Converting to Logarithmic Form

You can convert from exponential to log form simply by memorizing the pattern. Whatever was in the exponent in the exponential form (in red) goes by itself, on the other side of the equals sign, in the logarithmic form. Whatever was by itself in the exponential form (in green), goes inside the log part (written to the right of the word 'log') in the logarithmic form.

Here is a way to remember it:

Pretend that you are a kid and you have been grounded to your room for the evening by your parents for attempting to divide by zero - which we all know could result in the universe imploding. You happen to be lucky enough to have an identical twin, so you devise a plan. You can sneak out of the house to attend the local Star Trek convention, but she will have to take your place. If you are to be 'free,' she will have to be 'locked up.' If you stay locked up, she can stay free. You both can't get out at once.

If an equation is in exponential form, whatever is in the exponent is 'locked up.' You can't really get to it or isolate it. What's on the other side of the equation, however, is 'free.' It is easy to manipulate. If the equation is in log form, however, the part inside the 'log' is locked up. It can't be isolated without switching back to exponential form.

Free vs locked up

In equation (1), for example, the y can be easily solved for by subtracting both sides by 5 and then dividing by 3. The y part is 'free.' The x, however, can't be easily solved for (without using logs) as it is locked up in the exponent.

Exponential form

If you change the equation to logarithmic form, the green part on the left (3y + 5) will trade its freedom for the freedom of the red part (-3x). Equation (2) is now in log form.

Log form

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