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.