The Laws of Logs

Logarithms are the inverse of exponentials. If $a^y = x$, then $\log_a(x) = y$. This means logarithms answer the question: "To what power must the base $a$ be raised to get $x$?"

Recap:

1. Base: The number $a$ is the base of the logarithm, and it must be $a > 0$ and $a \neq 1$.
2. Argument: The value $x$ inside the logarithm ($\log_a(x)$) is called the argument and must satisfy $x > 0$.

---

The following are the key laws of logarithms, which help simplify expressions and solve equations involving logs:

3. Product Rule: $\log_a(xy) = \log_a(x) + \log_a(y)$

This means the logarithm of a product equals the sum of the logarithms of the factors.

4. Quotient Rule: $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator.

5. Power Rule: $\log_a(x^q) = q \cdot \log_a(x)$

If the argument has an exponent, you can bring the exponent in front as a multiplier.

6. Negative Argument Rule: $\log_a\left(\frac{1}{x}\right) = -\log_a(x)$

The logarithm of the reciprocal of a number equals the negative logarithm of the number.

7. Logarithm of 1: $\log_a(1) = 0$

This is because $a^0 = 1$ for any base $a$.

8. Logarithm Identity: $a^{\log_a(x)} = x$

Raising the base aa to the logarithm of $x$ returns $x$.

9. Inverse Property: $\log_a(a^x) = x$

The logarithm of a base raised to a power gives the power itself.

---

$$\log_a(xy)=\log_ax+\log_ay$$ $$\log_a\left(x^q\right)=q\log_ax$$ $$\log_a\left(\tfrac{1}{x}\right)=-\log_ax$$ $$a^{\log_ax}=x$$ $$\log_a\left(\tfrac{x}{y}\right)=\log_ax-\log_ay$$ $$\log_a1=0$$ $$\log_a\left(a^x\right)=x$$

Example

Summary:

• Logarithms are the inverse of exponents.
• These laws are crucial for solving logarithmic and exponential equations.