### Assignments:

Unfinished Assignment Study Questions for Lesson 27

### 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.