Chapter 4, Problem 4RCC

a.

To determine

How is the logarithmic function $y={\log }_{a}x$ defined?

Answer to Problem 4RCC

  $\boxed{a^y=x}$

Explanation of Solution

Given information:

How is the logarithmic function $y={\log }_{a}x$ defined?

Calculation:

The inverse of an exponential function is called the logarithm function. We defined the logarithm function $y={\log }_{a}x$ where $a\ne 1$ is equal to

  $a^y=x$

Hence, the solution is $\boxed{a^y=x}$

b.

To determine

What is the domain of this function $y={\log }_{a}x$ ?

Answer to Problem 4RCC

  $\left(0,\infty \right)$

Explanation of Solution

Given information:

What is the domain of this function $y={\log }_{a}x$ ?

Calculation:

The inverse of an exponential function is called the logarithm function. We can determine its domain by looking at the range of the exponential function, range of any exponent is the interval:

  $\left(0,\infty \right)$

Thus, conclusion is domain $\left(0,\infty \right)$

Hence, the solution is $\left(0,\infty \right)$

c.

To determine

What is the range of this function $y={\log }_{a}x$ ?

Answer to Problem 4RCC

All real numbers

Explanation of Solution

Given information:

What is the range of this function $y={\log }_{a}x$ ?

Calculation:

The inverse of an exponential function is called the logarithm function. The range of logarithm function will be equal to domain of logarithm function.

Since the domain of logarithm function is all real numbers.

Hence, the solution is all real numbers

d.

To determine

Sketch the general shape of the graph of the function.

Answer to Problem 4RCC

The solution is shown in graph.

Explanation of Solution

Given information:

Sketch the general shape of the graph of the function $y={\log }_{a}x,if,a>1$

Calculation:

The graph of function $y={\log }_{a}x$ with base $a=2$ .

  .

Hence, the solution is shown in graph.

e.

To determine

What is the natural logarithm?

Answer to Problem 4RCC

Logarithm function with base $e$

Explanation of Solution

Given information:

What is the natural logarithm?

Calculation:

Take any positive number that is not equal to $1$ to be the base of logarithm function.

So, of logarithm function with base $e$ is known as natural logarithm.

Hence, the solution is Logarithm function with base $e$

f.

To determine

What is the common logarithm?

Answer to Problem 4RCC

Logarithm function with base $10$

Explanation of Solution

Given information:

What is the common logarithm?

Calculation:

Take any positive number that is not equal to $1$ to be the base of logarithm function.

So, of logarithm function with base $10$ is known as common logarithm.

The common logarithm is denoted by leaving off the base as it is assumed to be $10$ .

Hence, the solution is Logarithm function with base $10$