Properties and Functions of Logarithms

The Product Rule states that $\log_b(MN) = \log_b(M) + \log_b(N)$. For example, $\log_2(2 \cdot 3) = \log_2(2) + \log_2(3)$ which simplifies to $1 + \log_2(3)$.

The significance of $\log_b(1)$ is that it equals 0 for any base $b$. This highlights that the logarithm of 1 does not change the value in multiplication.

For bases greater than 1, the graph of a logarithmic function increases and has a vertical asymptote at $x = 0$. It approaches the y-axis but never touches it.

<p>Using the Change of Base Formula: $\log_5(25) = \frac{\log_{10}(25)}{\log_{10}(5)}$. Since $25 = 5^2$, it equals $\frac{2 \cdot \log_{10}(5)}{\log_{10}(5)} = 2$.</p> Signup and view all the answers

<p>The domain of $f(x) = \log_b(x)$ is $x &gt; 0$, and the range is all real numbers ($-\infty &lt; f(x) &lt; \infty$). This is important because it defines where the function is valid and how it behaves.</p> Signup and view all the answers

<p>First, isolate the logarithm, then convert to exponential form: $x = 2^3$, which gives $x = 8$. Finally, check that $x = 8$ is in the domain.</p> Signup and view all the answers

<p>The Quotient Rule states that $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$. For example, $\log_2\left(\frac{8}{2}\right) = \log_2(8) - \log_2(2)$ simplifies to $3 - 1 = 2$.</p> Signup and view all the answers

<p>The Power Rule states that $\log_b(M^p) = p \cdot \log_b(M)$. Since $9 = 3^2$, we have $\log_3(9) = \log_3(3^2) = 2 \cdot \log_3(3) = 2$.</p> Signup and view all the answers

<p>If $b &gt; 1$, the function increases; if $0 &lt; b &lt; 1$, it decreases. This difference affects the overall shape of the graph and its applications.</p> Signup and view all the answers