How do you solve logarithmic equations?

Definition

A logarithmic equation is an equation that involves a logarithm with a variable inside its argument. To solve logarithmic equations, we often use properties of logarithms to combine terms and then rewrite the equation in exponential form.

Key properties:

• $log_b(MN) = log_b M + log_b N$
• $log_bleft(\frac{M}{N}\right) = log_b M - log_b N$
• $log_b(M^k) = k log_b M$

Worked Example

Solve:
$log_2(x + 1) = 3$

Step 1: Rewrite in exponential form

Recall that $log_b a = c$ means $a = b^c$.

So,
$$ x + 1 = 2^3 $$

Step 2: Simplify

$$ x + 1 = 8 $$

Step 3: Solve for $x$

$$ x = 8 - 1 = 7 $$

Step 4: Check the solution

Plug $x = 7$ back into the original equation:

$log_2(7 + 1) = log_2(8) = 3$ (since $2^3 = 8$)

The solution is valid.

Takeaways

• Use logarithm properties to combine or simplify terms.
• Convert the logarithmic equation to exponential form to solve for the variable.
• Always check your solution to ensure the argument of the logarithm is positive.