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Chapter 10: Problem 6

Write in logarithmic form. $$3^{6}=729$$

Short Answer

Expert verified
$$\text{log}_3{729} = 6$$

Step by step solution

01

- Identify the Form

The given equation is in the exponential form: $$a^b = c$$ Where: $$a = 3$$ $$b = 6$$ $$c = 729$$
02

- Apply Logarithmic Definition

The relationship between the exponential form and logarithmic form is given by: If $$a^b = c$$ then $$\text{log}_a{c} = b$$
03

- Convert to Logarithmic Form

Using the identified values in step 1 and applying the logarithmic definition from step 2, the equation $$3^6 = 729$$ can be written in logarithmic form as: $$\text{log}_3{729} = 6$$

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Key Concepts

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

Exponential Form
Exponential form is a way of expressing numbers as a base raised to an exponent. For instance, in the equation $$3^6 = 729$$, we have a base of 3 and an exponent of 6. In general, an exponential form looks like this: $$a^b = c$$ where:
  • **a** is the base
  • **b** is the exponent
  • **c** is the result
You can think of it as multiplying the base a, b times. So, $$3^6$$ means multiplying 3 by itself 6 times:
3 * 3 * 3 * 3 * 3 * 3 = 729.
Logarithms
Logarithms are the inverse operation to exponentiation. In simple terms, a logarithm explains how many times you need to multiply the base number to get another number.
For example, if we take the equation $$3^6 = 729$$, a logarithm will help us determine the exponent. The logarithm form of this equation would be $$\text{log}_3{729} = 6$$, which states that 3 should be multiplied by itself 6 times to get 729.
Basic properties of logarithms include:
  • $$\text{log}_a(a) = 1$$ because any number raised to the power of 1 is itself
  • $$\text{log}_a(1) = 0$$ because any number raised to the power of 0 is 1
Logarithmic Definition
The logarithmic definition relates exponential and logarithmic forms. If we know that exponentiation has the general form: $$a^b = c$$,
then its logarithmic form is: $$\text{log}_a(c) = b$$.
This is just another way to say that.logarithms provide us with the exponent when the base and the result are known.
So for the example we鈥檙e working with: $$3^6 = 729$$,
Applying the logarithmic definition, we write it as: $$\text{log}_3{729} = 6$$. This directly translates to: 鈥�3 raised to what power equals 729?鈥� The answer is 6.
Base of Logarithm
The base of a logarithm is the number that is raised to a power. In logarithmic form, the base is denoted as the subscript next to the log symbol.
For instance, in $$\text{log}_3{729} = 6$$, the base is 3.
Here are a few things to remember about bases:
  • The base must be a positive number and not equal to 1
  • Common bases are 10 (common logarithm) and e (natural logarithm)
In our equation, $$3^6 = 729$$, converting to logarithmic form gives us: $$\text{log}_3{729} = 6$$, with 3 serving as the base. It tells us what power we need to raise 3 to get 729.
Understanding the base is crucial for solving more complex logarithmic problems.

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Most popular questions from this chapter

Solve each equation. Give exact solutions. $$ \log _{5}(12 x-8)=3 $$

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Solve each problem. Suppose that \(\$ 3000\) is deposited at \(3.5 \%\) compounded quarterly. (a) How much money will be in the account at the end of 7 yr? (Assume no withdrawals are made.) (b) To one decimal place, how long will it take for the account to grow to \(\$ 5000 ?\)

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