How to solve the equation log4(x + 20) = 3

Step 1: Understand the logarithmic equation

The general form of a logarithmic equation is:

This can be rewritten in its exponential form as:

Step 2: Convert the logarithmic equation to its exponential form

Given our equation $\log_4(x + 20) = 3$, we can rewrite it as:

Step 3: Simplify the exponential expression

Calculate $4^3$:

So the equation becomes:

Step 4: Solve for $x$

Subtract 20 from both sides:

Conclusion

The solution to the equation is:

You can verify the solution by substituting $x$ back into the original equation:

Exponential Form Conversion

In mathematics, exponential form is a way to express numbers as powers of a base. This is particularly useful for simplifying expressions, performing calculations, and solving equations involving exponentials.

Basic Concept

The exponential form of a number is written as:

Here, $a$ is the base and $b$ is the exponent or power.

Converting to Exponential Form

To convert a number to exponential form, express it as a product of repeated factors of the base.

For example, the number 1000 can be expressed in exponential form. Since $1000 = 10 \times 10 \times 10$, it can be written as:

Examples

1. Convert 81 to exponential form:

   • Identify the base and the exponent.
   • Since $81 = 3 \times 3 \times 3 \times 3$, we can write: $$81 = 3^4$$

2. Convert $\frac{1}{64}$ to exponential form:

   • Recognize that $64 = 2^6$.
   • Since we are dealing with a reciprocal, we write: $$\frac{1}{64} = 2^{-6}$$

Exponential Form in Algebra

Exponential form is used to simplify algebraic expressions. For example, consider the expression:

Using the properties of exponents, we can simplify this to:

Logarithmic Connection

There is a strong connection between logarithmic and exponential forms. If you have:

You can express this relationship in logarithmic form as:

Summary

• Exponential form is a way to express numbers as powers of a base.
• To convert a number to this form, find the base and write the number as a product of repeated factors.
• This form is particularly useful in algebra for simplifying expressions and solving equations.

Understanding exponential form conversion is crucial for many areas of mathematics, from basic arithmetic to advanced calculus.

Importance of Logarithm Properties

Understanding logarithm properties is crucial because they simplify complex mathematical expressions, making calculations more manageable. Logarithms are particularly useful in fields such as science, engineering, and finance, where exponential growth or decay processes are common.

Key Logarithm Properties

1. Product Property: This property states that the logarithm of a product is the sum of the logarithms of the factors:

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

   This property is helpful when you need to break down products into simpler summations.

2. Quotient Property: According to this property, the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator:

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

   It's useful for dividing terms in a logarithmic form.

3. Power Property: This states that the logarithm of a number raised to a power is the power multiplied by the logarithm of the base number:

   $$\log_b (x^y) = y \cdot \log_b (x)$$

   This is particularly helpful for dealing with exponential terms.

4. Change of Base Formula: This formula allows you to change the base of a logarithm, making it easier to compute with different bases:

   $$\log_b (x) = \frac{\log_k (x)}{\log_k (b)}$$

   Here, $k$ can be any positive number (commonly $10$ or $e$).

5. Logarithm of 1: The logarithm of 1 to any base is always 0:

   $$\log_b (1) = 0$$

   This is a readily applicable property in simplifying expressions.

Applications

• Simplifying Calculations: By breaking down complex expressions, these properties make it easier to calculate logarithmic values manually or using computers.
• Solving Equations: They help convert exponential equations, which are often difficult to solve, into linear ones.
• Analyzing Growth/Decay: Logarithms are essential in interpreting data related to exponential growth or decay, such as population growth, radioactive decay, and interest calculations.

By mastering these key logarithm properties, you can handle a wide range of mathematical problems more effectively and efficiently.