Logarithmic Equations and Properties Quiz

Logarithm

A logarithm is the inverse operation of exponentiation. It is a mathematical function that calculates the exponent value required to obtain a specific value. In other words, it is a way of expressing the power to which a base must be raised to produce a given value. The base can be any positive number, although the most common base is 10, and the result is called a logarithm of base 10 or common logarithm.

Solving Logarithmic Equations

To solve a logarithmic equation, we need to isolate the logarithm and then use the inverse operation of exponentiation to solve for the variable. For example, to solve the equation:

$$10^x = 100$$

We can take the logarithm of both sides with base 10:

$$\log_{10}(10^x) = \log_{10}(100)$$

Using the property that $\log_{a}(a^x) = x$ for any base $a$, we can simplify the equation:

$$x$$

The solution to the equation is:

$$x = 2$$

This means that $10^2 = 100$, so the value of $x$ is 2.

Properties of Logarithms

There are several properties of logarithms that are useful in solving logarithmic equations:

1. Power Property: $\log_{a}(a^x) = x$ for any base $a$ and any positive value of $x$.

2. Product Property: $\log_{a}(P \times Q) = \log_{a}(P) + \log_{a}(Q)$ where $a$, $P$, and $Q$ are positive numbers.

3. Quotient Property: $\log_{a}(P / Q) = \log_{a}(P) - \log_{a}(Q)$ where $a$, $P$, and $Q$ are positive numbers.

4. Power of a power Property: $\log_{a}(a^{x^y}) = xy$ where $a$ is a positive number, $x$ and $y$ are any real numbers.

5. Logarithm of 1 Property: $\log_{a}(1) = 0$ for any base $a$ greater than 0.

6. Logarithm of zero Property: $\log_{a}(0)$ is undefined for any base $a$ greater than 0.

7. Change of base formula: $$\log_{b}(x) = \frac{\log_{a}(x)}{\log_{a}(b)}$$ where $a$, $b$, and $x$ are positive numbers.

These properties can be used to simplify logarithmic expressions and solve logarithmic equations.