Algebra II: Basics of Logarithms and Exponents

Before handheld calculators, students used tables of logarithms (or logs) to do calculations in physics and other science classes. Those tables of logarithms allowed you to do multiplication or division problems such as 456,000,000,000 × 892,658,000,000 or 0.00000045873 ÷ 0.0000135 by simply adding or subtracting numbers from the table. What were those numbers? They were the exponents you put on a 10 to get that particular number.

Why exponents? Because when you multiply numbers with the same base, you add exponents, and when you divide numbers with the same base, you subtract exponents. Here's a quick example:

Multiply 125 × 8.

Yes, you can do that quickly by hand and get 125 × 8 = 1,000. Using a table of logarithms, you'd find that 125 = 102.09691 and 8 = 100.90309. Add the two exponents together, and you have 2.09691 + 0.90309 = 3. And what power of 10 has an exponent of 3? Why 1,000, of course. Not all problems come out so conveniently, but this example still shows why logarithms work so well to do multiplication and division of very large and very small numbers.

The laws of logarithms are usually used to help you solve logarithmic equations. Why solve logarithmic equations? Because so many of the sciences use formulas and involve computations that require working with logarithmic and exponential expressions.

Exponential functions and logarithmic functions are closely related. The inverse of an exponential function is a logarithmic function, and vice versa. Depending on what you're doing/computing, one form or the other works best. Being able to quickly change the equation e–0.3x = 4 to –0.3x = ln(4) allows you to solve for the variable x with relative ease. The "ln" in the equation is, of course, a logarithm in base e.

Here are the basic relationships and rules involving logarithms and exponents:

  • Base 10: log10(x) = log(x)

    These are the common logarithms. When you don't see a subscript 10 after the "log," you assume the base is 10.

    On scientific calculators, the "log" button is used for these common logarithms.

    Equivalence:

    image0.png
  • Base e: loge(x) = ln(x)

    These are the natural logarithms. When you see "ln," you assume the base is e. The value of e is approximately 2.71828.

    On scientific calculators, the "ln" button is used for these natural logarithms.

    Equivalence:

    image1.png
  • Laws of logarithms: All the following laws are given in terms of "log" but apply to natural logs, too:

    image2.png
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