Watching this resources will notify you when proposed changes or new versions are created so you can keep track of improvements that have been made.
Favoriting this resource allows you to save it in the “My Resources” tab of your account. There, you can easily access this resource later when you’re ready to customize it or assign it to your students.
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 10 10 10 = 103.
More generally, if x = by, then y is the logarithm of x to base b, and is written y = logb(x), so log10(1000) = 3 .
The logarithm of a product is the sum of the logarithms of the factors:
$log_b(xy) = log_bx + log_by$
Similarly, the logarithm of the ratio of two numbers is the difference of the logarithms.
$log_x(a/b) = log_xa - log_xb$
To prove this, let m = logxa.
Rewrite the above expression as an exponent.
xm = a (logxa asks " x to what power is a ?"
And the equation answers: "x to the m is a.")
Let n = logxb.
If we replace a and b based on the previous equations, we get:
$log_x(a/b) = log_x(x^m/x^n)$
This can be further simplified to:
$log_x(x^m/x^n) = log_x(x^m-^n)$
Which, using the first law of exponents, can be written as:
$log_x(x^m/x^n) = m-n$
This is the key step.
Therefore, it can be seen that the properties of logarithms come directly from the laws of exponents.
Replacing m and n with what they were originally defined as results in this equation: