# Logarithm Function: Properties and Solutions [Attached Image]

• Victim
In summary, the conversation revolves around simplifying the expression ##\log _3 \left( a^{\log _3 7} \right)##, with the hint given to use the property ##a^{(log_m n)}=n^{(log_m a)}##. The attempt at a solution involves solving for a, b, and c with the values of 20, 38, and √(11)-25 respectively, but it is stated that they are not necessary to solve the problem. A different hint is given to help with the simplification process.

## Homework Statement

I have attached image of question.[/B]

## Homework Equations

all the properties of log
a^(logₘn)=n^(logₘa)[/B]

## The Attempt at a Solution

in the attached image
[/B]

#### Attachments

• 1534508914003-822972847.jpg
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##\log _3 \left( a^{\log _3 7} \right)## is of the form ##\log _3 \left( a^y \right)##, Can you simplify this further?

George Jones said:
##\log _3 \left( a^{\log _3 7} \right)## is of the form ##\log _3 \left( a^y \right)##, Can you simplify this further?
and then

Victim said:
and then
You tell us, George gave you a hint.

jedishrfu said:
You tell us, George gave you a hint.
OK by this method I got a=20 b=38 c=√(11)-25
Then they are in power of log₃7,log₇11 and log₁₁25.How to solve these powers

Victim said:
OK by this method I got a=20 b=38 c=√(11)-25
Then they are in power of log₃7,log₇11 and log₁₁25.How to solve these powers
The problem states that a, b, and c are positive. The value you give for c, ##\ \sqrt{11\,}-25\,,\ ## is negative.

Moreover, as it turns out, a, b, and c are all irrational. Fortunately, you do not need to solve for any of them to solve this problem.

Here is a hint that's different than the one given by George Jones.
Notice that ##\displaystyle \ X^{Y^ 2}=X^{Y\cdot Y}=\left(X^Y\right)^Y \ ##​
.

## 1. What is a logarithm function?

A logarithm function is the inverse of an exponential function. It helps us solve for the exponent in an exponential equation. In other words, it tells us what power we need to raise a certain base to in order to get a given number.

## 2. What are the properties of logarithm functions?

There are several properties of logarithm functions, including the product rule, quotient rule, power rule, and change of base rule. These properties allow us to manipulate logarithmic expressions and solve for unknown variables.

## 3. How do you graph a logarithm function?

To graph a logarithm function, you first need to determine the vertical and horizontal asymptotes. Then, you can plot points by choosing a base value, plugging it into the logarithmic equation, and solving for the corresponding y-value. Finally, you can connect the points to create a curve.

## 4. What is the domain and range of a logarithm function?

The domain of a logarithm function is all positive real numbers, since we cannot take the logarithm of a negative number. The range is all real numbers, since the output of a logarithm function can be any real number depending on the input.

## 5. How are logarithm functions used in real life?

Logarithm functions are used in a variety of fields, including finance, biology, and physics. In finance, they can be used to calculate compound interest. In biology, they can be used to measure the pH scale. In physics, they can be used to calculate the intensity of sound or earthquakes.