If you were to look at the exponential equation \({5}^{x}=125\), you would easily be able to say that \(x=3\) because we know that \(5\) raised to the third power is \(125\). However, if we look at the equation \({5}^{x}=95\), we wouldn’t be able to come up with an exact answer immediately. We would know that \(x\) has to be between \(2\) and \(3\) but would only be approximating from there. We are now going to learn about logarithms which will help us quickly solve this type of equation.
A logarithm is the power that a base number must be raised to in order to be equal to another number. Let’s go back to our example of \({5}^{x}=125\). If we let \(x=3\), then we could write it as \({5}^{3}=125\). The power that we need to raise \(5%\) to in order to equal \(125\) is \(3\) and, using logarithms, we can express that as \(\log_5125=3\).
A logarithm is the power that a base number must be raised to in order to be equal to another number. Let’s go back to our example of \({5}^{x}=125\). If we let \(x=3\), then we could write it as \({5}^{3}=125\). The power that we need to raise \(5%\) to in order to equal \(125\) is \(3\) and, using logarithms, we can express that as \(\log_5125=3\).
Let \(b\) and \(y\) be positive numbers where \(b\ne1\). The exponential equation \(b^x=y\) is equivalent to the logarithmic equation \(\log_by=x\). Where \(b\) is the base, \(y\) is called the argument and \(x\) is called the logarithm.
Example 1: Rewrite each logarithm in exponential form.
1) \(\log_381=4\) 2) \(\log_{\frac{1}{5}}25=-2\) Rewrite each in logarithmic form. 1) \(4^3=64\) 2) \(\left(\frac{2}{3}\right)^{^{-2}}=\frac{9}{4}\) |
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We will work with several different properties of logarithms, but here are a few to get you started.
- \(\log_bb=1\) (because any value of \(b\) raised to the power of \(1\) is that same number)
- \(\log_b1=0\) (because any value of \(b\) raised to the power of \(0\) is \(1\))
- \(\log_bb^x=x\) (because any value of \(b\) raised to the power of \(x\) is \(b^x\))
Example 2: Evaluate each log expression:
1) \(\log_28\) 2) \(\log_71\) 3) \(\log_{\frac{1}{16}}32\) 4) \(\log\left(\frac{1}{1000}\right)\)
There are two logarithms that are used often and that your calculator can evaluate very quickly. A common logarithm is a logarithm with a base of \(10\) and is written as \(\log x\) (note that there is no base written…it is assumed to be \(10\)). A natural logarithm is a logarithm with a base of \(e\) and is written as \(\ln x\) (note that there is no base written…it is assumed to be \(e\)).
We are now going to look at some more properties of logarithms which will be helpful in solving equations involving logarithms.
- Product Property: \(\log_b\left(mn\right)=\log_bm+\log_bn\)
- Quotienty Property: \(\log_b\left(\frac{m}{n}\right)=\log_bm-\log_bn\)
- Power Property: \(\log_bm^n=n\log_bm\)
Example 3: Expand \(\log_3\left(\large\frac{4y^5}{27x}\right)\).
Example 4: Condense \(\ln5-3\ln x-\ln25\).
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