# Laws of Logs

## The Importance of Knowing and Implementing the Laws of Logs

The laws of logarithms are crucial in simplifying and solving complex logarithmic equations. By following these essential rules, you can manipulate logarithms for easier solving. It is crucial to ensure consistency in bases when working with logarithms.

### The Four Main Laws of Logs

• Product Law: When the bases of both logarithms are the same, they can be rewritten as an exponential function.
• Quotient Law: If the bases are identical, the logarithms can be rewritten as an exponential function.
• Change of Base: This law can be found in the formula booklet provided during exams. It allows you to convert logarithms to exponential functions.
• Power Law: This law states that when raising a logarithm to a power, you can rewrite it as the product of that power and the logarithm.

Note: Other laws, such as the Reciprocal Law, Log of the Base, and Log of 1, are more specialized and should only be used in specific situations.

### Product Law in Action

1. If both logarithms have the same base, rewrite them as exponential functions.
2. Use the exponential (indices) rule to solve.
3. Take the log of both sides.
4. The exponential and logarithm with the same base will cancel out.
5. Therefore, the final result depends on the values defined in the logarithms.

### Quotient Law Explained

1. If the bases are the same, rewrite the logarithms as exponential functions.
2. Solve the exponential functions separately.
3. Take the log of both sides.
4. The exponential and logarithm with the same base will cancel out.
5. The final result is determined by the values of c and d in the logarithms.

### Change of Base Demystified

1. Let x be the solution for the exponential function with base a and exponent k. Rewrite it as an exponential function.
2. Take the log of both sides.
3. Use the power rule to simplify the equation.
4. Substitute the value of k back into the equation.
5. Rearrange the equation to isolate k.
6. Since k is known, you can substitute it back into the equation.

### Reciprocal Law Unlocked

The Reciprocal Law, written as a combination of exponential rules with negatives, allows for the use of the power log rule to simplify equations.

### Log of the Base Simplified

When dealing with logarithms involving the log of a base, convert them to exponential functions and solve like a regular equation.

### Simplifying and Solving with Logarithms

Now, let's see how the different laws can be applied with some examples of simplifying and solving logarithms.

#### Using the Product Law

Simplify and solve: Log(8) + Log(2) = Log(16)

Log(8) + Log(2) = Log(8 x 2) = Log(16)

#### Using the Quotient Law

Solve: Log(25) - Log(5)

Log(25) - Log(5) = Log(25/5) = Log(5) = 0.699 (3 sf)

#### Simplifying Multiple Log Laws

It is helpful to simplify individual logarithms before using multiple log laws.
Simplify: Log(81) - Log(9)

Log(81) - Log(9) = Log(81/9) = Log(9) = 0.698 (3 sf)

### Proving the Laws of Logs

While not necessary, it is important to understand each step and why it occurs when proving logarithm laws during exams.
1. Using the Power Law,
2. Using the Quotient Law,
3. The logarithm must be converted into an exponential function when removing it.

## Key Takeaways from the Laws of Logs

The four main laws of logs - Product Law, Quotient Law, Change of Base, and Power Rule - are crucial in manipulating logarithms. Other laws, such as the Reciprocal Law, Log of the Base, and Log of 1, are specialized and should only be used in specific contexts.

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