### Lesson Objectives:

- The Product Rule- The Power Rule

- The Quotient Rule

- The Logarithm of a Base to a Power

- A Base to a Logarithmic Power

The Product Rule:

The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.

Let m and n be any two numbers, and x and y their logarithms. Then by definition, `10^x = m`, and `x = log m`. And `10^y = n`, and `y = log n`.

Multiplying `10^x*10^y = m*n`, or `10^(x+y) = m*n`, because we can add their exponents. So the `log(m*n) = x + y`, because we converted this to log form.

And now substitute in your x and your y-values, `log m` and `log n`, so that we get the product rule, which is `log(m*n) = log m + log n`.

Express as a single logarithm and if possible, simplify.

`log_a50+log_a2`. So because of the Product Rule, we can rewrite his as `log_a(50*2)`, which is `log_a(100)`.

The Power Rule:

The logarithm of any power of a number is equal to the logarithm of the quantity multiplied by the exponent of the power.

Let m be any number, and x its logarithm. Then, by definition, `10^x = m`, and `log m = x`.

So if we raise both members to a power denoted by p, then we get `10^(p*x) = m^p`, or if we convert this to log form, we have `log m^p = p*x`. And then substitute `log m` in for x, and we get `p*log m`.

So the Power Rule is `log m^p = p*log m`.

Express as a product.

`log_ak^(-3)`. So by the Power Rule, we can simplify this to `-3*log_ak`.

The Quotient Rule:

The logarithm of the quotient of two numbers is equal to the logarithm of the dividend minus the logarithm of the divisor.

Let `m` and `n` be any two numbers, and `x` and `y` their logarithms. Then by definition, `10^x = m`, and `log m = x`. `10^y = n`, and `log n = y`.

So if we divide `(10^x)/(10^y)`, this is equal to `m/n`, or `10^(x-y) = m/n`. And if we convert this to log form, this is the same as `log(m/n) = x - y`.

Now substitute the values of `x` and `y`, `log m` and `log n`, and you have `log(m/n) = log m - log n`, which is the Quotient Rule.

Express as a single logarithm and simplify:

`log(1,000,000)-log(1,000)`. By the Quotient Rule, we can rewrite this as `log(1,000,000/1,000)`, and then we can simplify by dividing the numerator and the denominator by 1,000. So we have `log 1,000`, which is the same as `log_(10)1000`, or `log_(10)10^3`, and by the power rule, this is the same as `3*log_(10)10`. `log_(10)10` is just 1, so this is the same as 3.

The Logarithm of a Base to a Power:

`log_aa^x = x` for any base, a, and any real number, x.

A Base to a Logarithmic Power:

`""_alog_ax = x` for any base, a, and any real number, x.

Simplify.

`""_5log5(3x)`. This can be simplified as 3x.