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Chapter 4: Problem 2

In Exercises 1鈥�8, write each equation in its equivalent exponential form. $$ 6=\log _{2} 64 $$

Short Answer

Expert verified
The equivalent exponential form of the equation is \(2^6 = 64\).

Step by step solution

01

Understand the formula

Logarithm \( \log_b a = x \) can be converted to exponential form \( b^x = a \). This is a logarithm property.
02

Identify the values

In the given equation \(6 = \log_2 64 \), the base (\(b\)) is 2, the logarithm of the number (\(a\)) is 64, and the result (\(x\)) is 6.
03

Apply the values to the formula

Using the identified values for \(b\), \(x\) and \(a\), the equivalent exponential form of the equation is \(2^6 = 64\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm
A logarithm tells us the power to which a base number must be raised to get another number. For instance, in the equation \(\log_2 64 = 6\), it indicates that the base, 2, must be raised to the power of 6 to result in 64. Understanding logarithms is essential because they help us unravel exponential expressions and solve equations where the unknown is an exponent.
Logarithms have a specific structure. A logarithm statement includes:
  • The base, which is the number being multiplied.
  • The argument, which is the result of the multiplication raised to the power of the unknown.
  • The logarithm itself, which reveals the exponent or power.
This structure helps in re-writing equations either in logarithmic or exponential form. Recognizing the conversion between these forms can simplify complex mathematical problems.
Exponentiation
Exponentiation involves multiplying a number by itself a specific number of times expressed as an exponent. This is what the conversion from a logarithm to its exponential form reveals. In the equation \(2^6 = 64\), 2 is the base, and 6 is the exponent, which signifies that we multiply 2 by itself six times. The process results in 64.
Exponentiation is a fundamental operation in algebra and beyond, forming the basis for understanding growth patterns, such as in finance, sciences, and technology. Here are some key points about exponentiation:
  • It can be used to express very large or tiny numbers efficiently.
  • The base can be any real number, but commonly, positive integers are used.
  • Exponents can be positive, negative, or even fractions.
Understanding how exponentiation works is crucial for solving equations in science and engineering.
Logarithm Properties
The properties of logarithms are powerful tools that make calculations and problem-solving more manageable. These properties allow us to break down complex logarithmic expressions using rules similar to those of multiplication and addition.
Some essential properties of logarithms include:
  • Product Property: \(\log_b (MN) = \log_b M + \log_b N\)
  • Quotient Property: \(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)
  • Power Property: \(\log_b (M^p) = p \cdot \log_b M\)
  • Change of Base Formula: \(\log_b a = \frac{\log_k a}{\log_k b}\)
These properties not only help in simplifying logarithmic expressions but also in solving exponential and logarithmic equations more efficiently. They are foundational for further studies in calculus, algebra, and applied mathematics.

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Most popular questions from this chapter

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$ 5^{x}=3 x+4 $$

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