Simplifying Logarithmic Expressions

Simplifying logarithmic expressions entails using various properties of logarithms. Most of them are analogous to the properties of exponents you're already familiar with. Let's get started!

Properties of logarithms useful when simplifying logarithmic expressions

It's best to examine these properties one-by-one:

1. ${\log }_{b}b=1$ for any value b

The reason for this is that $b^1=b$ regardless of the value of b.

2. ${\log }_{b}1=0$ for any value b

The reason for this is that $b^0=1$ regardless of the value of b (undefined when $b=0$ ).

3. ${\log }_{b}0$ is undefined for all values of b

There is no x value for which $b^x=0$ .

4. ${\log }_{b}x$ is undefined whenever b is negative

At first glance, you might be thinking this one is wrong since ${\log }_2\left(-8\right)$ should equal 3 because ${\left(-2\right)}^3=-8$ , but remember that math rules must be applicable in all cases. Something like ${\log }_{-2}\left(-4\right)$ doesn't mean anything because the equation ${\left(-2\right)}^x=-4$ has no real solution. With no clear way to differentiate the two examples, we say that $\log \left(x\right)$ is undefined whenever x is a negative number.

5. ${\log }_b\left(xy\right)={\log }_b\left(x\right)+{\log }_b\left(y\right)$

This works because $b^m\times b^n=b^{(m+n)}$ . This is probably the logarithmic property you'll get the most use out of when simplifying logarithmic expressions.

6. ${\log }_b\left(\frac{x}{y}\right)={\log }_b\left(x\right)-{\log }_b\left(y\right)$

This is simply the inverse of the fifth property and will also be very useful when simplifying logarithmic expressions.

Practice questions on simplifying logarithmic expressions

a. Simplify the logarithmic expression: ${\log }_b\left(347x\right)$

The 347 looks like a big scary number, but the quantity involved is irrelevant. Using the rule:

${\log }_b\left(xy\right)={\log }_b\left(x\right)+{\log }_b\left(y\right)$

we can break the expression into:

${\log }_{}\left(347\right)+{\log }_{}\left(x\right)$

Then isolate the variable for further work. It should also be noted that we're working with logarithms in base 10 here since there is no subscript.

b. Simplify the logarithmic expression:

$\ln \left(\frac{2}{3}\right)$

The ln means we're working with the natural logarithm, but all of the properties of logarithms still apply. Using the rule ${\log }_b\left(\frac{x}{y}\right)={\log }_b\left(x\right)-{\log }_b\left(y\right)$ The expression becomes:

$\ln \left(2\right)-\ln \left(3\right)$

We could work that out with a calculator if we wanted to, but for now, there's no need.

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