91Ó°ÊÓ

To start with, let’s discuss the Power Rule of logarithms. This rule is essential when dealing with logarithms that have coefficients. The Power Rule states that: \ \( a \log(b) = \log(b^a) \)
This means you can take a coefficient in front of a logarithm and turn it into an exponent on the inside.
For example:
- If you have \( 15\log(c) \), you apply the power rule to get \( \log(c^{15}) \).
- Similarly, \( \frac{1}{4}\log(d) \) becomes \( \log(d^{\frac{1}{4}}) \).
This rule helps to simplify expressions significantly, especially when combining multiple logarithms later.

Next, let’s look at the Logarithm Subtraction Rule. This comes in handy when you’re subtracting logarithms. The rule is given by: \ \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \)
You can combine two logarithms by turning the subtraction into a division inside the logarithm.
For example:
- If you have the term \( \log(c^{15}) - \log(d^{\frac{1}{4}}) \), you can combine them into \( \log\left(\frac{c^{15}}{d^{\frac{1}{4}}}\right) \).
This rule allows you to deal with multiple logarithmic terms in a more manageable form, making it easier to simplify further.

When simplifying logarithmic expressions, it’s essential to combine and reduce them as much as possible. The steps involve:
- Applying the power rule to move coefficients into the logarithm.
- Using addition and subtraction rules to combine into a single logarithm.
For example, starting with \( 15 \log(c) - \frac{1}{4}\log(d) - \frac{3}{4} \log(k) \):
- Use the power rule to get \( \log(c^{15}) - \log(d^{\frac{1}{4}}) - \log(k^{\frac{3}{4}}) \).
- Apply the subtraction rule step-by-step to get a final, simplified form: \( \log\left( \frac{c^{15}}{d^{\frac{1}{4}} k^{\frac{3}{4}}} \right) \).
This makes the logarithmic expression easy to understand and use in further calculations.

Understanding exponents is crucial when working with logarithms. Exponents are used to express repeated multiplication of a base number.
For example, in the expression \( c^{15} \, c \) is the base and \ 15 \ is the exponent. This means \ c \ multiplied by itself 15 times.
Similarly, fractional exponents represent roots: \( d^{\frac{1}{4}} \) represents the fourth root of \ d \.
When simplifying within logarithms, keep in mind:
- Whole number exponents remain straightforward multiplications.
- Fractional exponents translate to roots, like squares or cubes.
Combining these insights lets you handle complex logarithmic and exponential expressions more effectively.