Integrated Math 3: Unlock Logarithmic Properties with These Solved Examples

Logarithmic Properties Explained with Examples 🧮

Here are several solved examples demonstrating the application of logarithmic properties. These examples are designed to help you master simplifying expressions and solving equations in Integrated Math 3.

1. Product Rule ➕

The product rule states: $log_b(MN) = log_b(M) + log_b(N)$

Example: Expand $log_2(8x)$

log_2(8x) = log_2(8) + log_2(x)
          = 3 + log_2(x)

2. Quotient Rule ➗

The quotient rule states: $log_b(\frac{M}{N}) = log_b(M) - log_b(N)$

Example: Expand $log_3(\frac{27}{y})$

log_3(\frac{27}{y}) = log_3(27) - log_3(y)
             = 3 - log_3(y)

3. Power Rule 🚀

The power rule states: $log_b(M^p) = p \cdot log_b(M)$

Example: Expand $log_5(25^x)$

log_5(25^x) = x \cdot log_5(25)
          = x \cdot 2
          = 2x

4. Change of Base Rule 🔄

The change of base rule states: $log_b(M) = \frac{log_c(M)}{log_c(b)}$

Example: Evaluate $log_4(20)$ using the change of base rule.

log_4(20) = \frac{log_{10}(20)}{log_{10}(4)}
         ≈ \frac{1.301}{0.602}
         ≈ 2.161

5. Combining Logarithmic Properties 🧩

Example: Expand $log(\frac{x^3\sqrt{y}}{z^2})$

log(\frac{x^3\sqrt{y}}{z^2}) = log(x^3\sqrt{y}) - log(z^2)
                     = log(x^3) + log(\sqrt{y}) - log(z^2)
                     = 3log(x) + \frac{1}{2}log(y) - 2log(z)

6. Solving Logarithmic Equations 🔑

Example: Solve $log_2(x) + log_2(x-2) = 3$

log_2(x) + log_2(x-2) = 3
log_2(x(x-2)) = 3
x(x-2) = 2^3
x^2 - 2x = 8
x^2 - 2x - 8 = 0
(x-4)(x+2) = 0
x = 4, x = -2

Since we cannot take the logarithm of a negative number, $x = -2$ is an extraneous solution. Therefore, $x = 4$.

7. Condensing Logarithmic Expressions 🗜️

Example: Condense $2log(x) + 3log(y) - log(z)$

2log(x) + 3log(y) - log(z) = log(x^2) + log(y^3) - log(z)
                          = log(x^2y^3) - log(z)
                          = log(\frac{x^2y^3}{z})

These examples should give you a solid foundation for understanding and applying logarithmic properties. Practice is key to mastering these concepts! 🚀