对数的加法（同底数）

Laws of Logs

Here's a summary of all the useful algebraic properties of logs that you can use!

Basic facts and identities:

1. $\log_{a}1=0$
2. $\log_{a}a=1$
3. $\log_{a}a^b=b$
4. $a^{\log_{a}b}=b$

Operations:

1. $\log_{a}bc=\log_{a}b+\log_{a}c$
2. $\log_{a}\frac{b}{c}=\log_{a}b-\log_{a}c$
3. $\log_{a}b^c=c\log_{a}b$, also known as log of a power
4. $\log_{a^b}c=\frac{1}{b}\log_{a}c$

Change of base:

1. $\log_{a}b=\frac{\log_{c}b}{\log_{c}a}$

Logarithm addition and subtraction

$$\log_ab+\log_ac=\log_abc$$ $$\log_ab-\log_ac=\log_a\frac{b}{c}$$

These are the properties of addition and subtraction for logarithms...

Let's think about where they come from.

Recall the relationship between exponents and logarithms: $$\log_xy=z \quad\text{if and only if}\quad x^z=y$$

We also know the following two properties of exponents: $$a^b\cdot a^c=a^{b+c} \quad \text{and} \quad \frac{a^b}{a^c}=a^{b-c}$$

Let's go one property at a time.

Addition

Suppose that $a^b=c$ and $a^d=e$. Let's look at a base $a$ logarithm of the product of $c$ and $d$: $$\log_a(ce)=\log_a(a^b\cdot a^d)$$ Since we know how to deal with products of exponents with the same base, we can rewrite this as: $$\log_a(a^b\cdot a^d)=\log_a(a^{b+d})$$ Now, by properties of logarithms, we can eliminate $a$: $$\log_a(a^{b+d})=b+d$$ But from our initial definitions, we know that $b=\log_ac$ and $d=\log_ae$, so: $$\log_a(ce)=\log_ac+\log_ae$$ And we have our formula!

Subtraction

Again, suppose that $a^b=c$ and $a^d=e$. Let's look at a base $a$ logarithm of the quotient of $c$ and $d$: $$\log_a{\frac{c}{e}}=\log_a{\frac{a^b}{a^d}}=\log_a(a^{b-d})=b-d=\log_ac-\log_ae$$ And we have our formula!

(Video) Properties of Logarithms

by yaymath

This video by yaymath explains the properties of logarithms and works through some practice problems.

$\large{\text{Summary}}$

There are three main log properties:

$$\begin{align}\log xy&=\log x+\log y\\ \log \frac{x}{y}&=\log x-\log y\\ \log x^n&=n\log x \\ \end{align}$$

These can be used to solve equations for logarithms. Let's work through a problem together.

Solve the following for $x$: $$3\log_5x - \log_54 = \log_516$$ First, we're going to condense our exponents. The first term, $3\log_5x$, can be rewritten with an exponent. $$\log5x^3 - \log_54 = \log_516$$ Now, we're going to translate the subtraction of our first two terms into division. $$\log_5\frac{x^3}{4} = \log_516$$ We have a log of the same base on both sides of the equation, so we can remove the logs and solve this the same way we'd solve any other equation. $$\begin{align} \frac{x^3}{4} &= 16\\ x^3&=64\\ x&=4 \end{align}$$

Try this question for yourself:

Let's work through this problem. First, we're going to deal with the exponents. Now we can combine our terms into division and multiplication, respectively. $$\log_3\frac{x^7}{4} = \log_316x^5$$ We have a log of the same base on each side, so we can ignore them and solve the same way we would any other equation. $$\begin{align} \frac{x^7}{4}&=16x^5\\ x^7&=16x^5 \cdot 4\\ x^7&=64x^5\\ \frac{x^7}{x^5}&=64\\ x^2&=64\\ x&=8 \end{align}$$

(Video) Properties of Logarithms

by PatrickJMT

This video by PatrickJMT covers addition and subtraction with logs of the same base.

$\large{\text{Summary}}$

The rules of logs state that: $$\log_a(xy)=\log_a(x)+\log_a(y)$$ $$\log_a\bigg(\frac{x}{y}\bigg)=\log_a(x)-\log_a(y)$$ $$\log_a(x^n)=n\log_a(x)$$

Let's expand a logarithmic expression to show this. $$\log_2\frac{x^3y}{z^2}$$ We can start by changing the fraction into subtraction. $$\log_2x^3y - \log_2z^2$$ Now we can expand the first term from multiplication into addition. $$\log_2x^3 + \log_2y - \log_2z^2$$ Finally, we can move the exponents to the front of each term. $$3\log_2x + \log_2y - 2\log_2z$$

Try a problem for yourself!

We can shorten our expression by multiplying the arguments $(2x$ and $8x)$ inside of the log. $$\begin{align}\log2x + \log8x &= \log(2x \cdot 8x)\\ &=\log(16x^2) \end{align}$$

(Video) Combining Logarithmic Expressions

by ThinkwellVids

This video by ThinkwellVids video goes over some more complex logarithm examples.

$\large{\text{Summary}}$

There are five main rules of logarithms that help with logarithm combination and arithmetic.

$$\begin{align}\log_bb &= 1\\ \log_b1 &= 0\\ \log_b(xy) &= \log_bx + \log_by\\ \log_b(\frac{x}{y}) &= \log_bx - \log_by\\ \log_b(x^y) &= y \log_bx \end{align}$$

Let's try a practice problem. We're going to combine the terms of the following: $$3\log_4x - 2\log_4z + 2\log_4w$$ First, we're going to turn the multiplication into exponents. $$\log_4x^3 - \log_4z^2 + \log_4w^2$$ Now, we're going to combine the two subtracted terms. $$\log_4(\frac{x^3}{z^2}) + \log_4w^2$$ The final step is to combine the added terms as multiplication. $$\log_4(\frac{x^3}{z^2}\cdot w^2)$$

Try a question for yourself:

Adding and Subtracting Logs (Same Base)