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Chapter 12: Problem 5

Write in logarithmic form \(4^{5}=1024\)

Short Answer

Expert verified
The logarithmic form is \( \log_4{1024} = 5 \).

Step by step solution

01

- Identify the components

Understand the components of the equation provided. The equation given is in exponential form, which is written as: y = b^x.Here, y = 1024, b = 4, and x = 5.
02

- Recall the logarithmic form

Recall that the logarithmic form of an equation is defined as: y = b^x can be written as \( \log_b{y} = x \).
03

- Substitute the components into the logarithmic form

Substitute the values from the exponential equation into the logarithmic form: \( \log_4{1024} = 5 \).

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Key Concepts

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

Exponential Form
The equation \(4^{5}=1024\) is written in its exponential form. But what does this mean? An exponential equation represents repeated multiplication. Here, the base is 4 and the exponent is 5. To break it down:
  • \textbf{Base (b)}: The number being multiplied (4).
  • \textbf{Exponent (x)}: The number of times the base is multiplied by itself (5).
  • \textbf{Result (y)}: The final value obtained after performing the multiplication (1024).

So, in simple terms, 4 is multiplied by itself 5 times to give 1024. This is \(4 \times 4 \times 4 \times 4 \times 4 = 1024\). Understanding this will help in converting it to logarithmic form.
Logarithms
Logarithms are another way to represent exponential equations. When we see an exponential equation like \(4^{5}=1024\), we can use logarithms to ask a different question: 鈥�4 raised to what power equals 1024?鈥�.

The notation used is \( \text{log}_b{y} = x \). In this,
  • \textbf{b} (base) remains the same (4).
  • \textbf{y} becomes the number we are interested in (1024).
  • \textbf{x} is the power or exponent (5).
So, the logarithmic equivalent of the equation \(4^{5}=1024\) is \( \text{log}_4{1024} = 5 \). Knowing this can help in algebraic manipulations, and solving equations become easier.
Equation Conversion
Converting between exponential and logarithmic forms is a key skill in algebra. Given the exponential form \(4^{5}=1024\), follow these steps to convert it to logarithmic form:
  • Identify the base (4), exponent (5), and result (1024).
  • Understand the general format: \(y = b^x \) converts to \( \text{log}_b{y} = x \).
  • Substitute the identified components into the logarithmic form.
The equation \(y = b^x\) thus transforms to \( \text{log}_4{1024} = 5 \). This method of conversion is useful, especially when dealing with larger numbers and complex equations. It helps in solving for unknowns and can simplify multiplication to addition.

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$\log _{a} m-\log _{a} n$$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 2^{19}$$

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{7} \frac{\sqrt[3]{13}}{p q^{2}}$$

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{3 x}=9 $$

$$ \left(\frac{4}{3}\right)^{x}=\frac{27}{64} $$
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