• $log_3(27)$ means "what power do I have to give $3$ to get $27$?"
• This is spoken as "log base $3$ of $27$." Thus,

If the base of a logarithm is not written, it is assumed to be $10$. This is a basic introduction to logarithms, which are used to find the powers needed to get a certain number from a base. The example uses base $3$ to find out which power of $3$ gives $27$.

$\log_a(y) = x \quad \text{if and only if} \quad a^x = y$

Here, $a > 0$ and $a \neq 1$ .

• Base 10 (Common Logarithm): $\log_{10}(y)$ is often written as $\log(y$).

$\log_{10}(y) \quad \text{or simply} \quad \log(y)$

• Base e (Natural Logarithm): $\log_e(y)$ is often written as $\ln(y)$, where $e \approx 2.718$.

$\log_e(y) \quad \text{or simply} \quad \ln(y) \quad \text{with} \quad e \approx 2.718$

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

$\log_a(xy) = \log_a(x) + \log_a(y)$

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

$\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

• Power Rule: $\log_a(x^n) = n \log_a(x)$

$\log_a(x^n) = n \log_a(x)$

• Change of Base Formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$, for any base $b$.

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

Question: Solve the equation $2\log_3(x) - \log_3(9) = 1$.

Solution:

1. Simplify the equation using logarithm properties:

$\log_3(9) = \log_3(3^2) = 2\log_3(3) = 2$

So the equation becomes:

$2\log_3(x) - 2 = 1$

2. Isolate the logarithm:

$2\log_3(x) = 3$

3. Solve for $\log_3(x)$:

$\log_3(x) = \frac{3}{2}$

4. Convert the logarithmic equation to an exponential equation:

$x = 3^{\frac{3}{2}} = \sqrt{3^3} = \sqrt{27} = 3\sqrt{3}$

Final Answer: $x = 3\sqrt{3}$.