# Solve Logarithmic Equation: [log_{2} 9/log_{2} 10] - log 3

• nvez
In summary, the conversation discusses the process of simplifying an expression involving logarithms, specifically changing the base and using the properties of logarithms to combine terms. The final answer is equal to log 3 and it is noted that it is important to understand the difference between equations and expressions in order to know which operations are allowed.
nvez

## Homework Statement

I have to take the following equation and make it one single log

$$[log_{2} 9/log_{2} 10] - log 3$$

The answer is equal to: $$log 3$$

I usually know how to do this however I have no clue in this case, I have the answer however.

## Homework Equations

None?
log x - log y = log (x/y)
log x + log y = log (x*y)

## The Attempt at a Solution

I have tried the putting what the divison part really is equal too then subtracting it or whatever but to no avail.

Try changing the base to log10 (I assumed that is what log was)

Thank you very much!

For future reference:

logc m = logs m / logs c

In this case:

log2 9 / log2 10 = log10 9

log 9 - log 3 = log (9/3) = log 3

Thanks again.

nvez said:

## Homework Statement

I have to take the following equation and make it one single log

$$[log_{2} 9/log_{2} 10] - log 3$$
A small point: this is not an equation. It's important to understand when you're working with an equation and when you're working with an expression, since the kinds of operations you can do are different. For example, if you're working with an equation, you can add a quantity to both sides, multiply both sides by a quantity, and so on, but you're very limited in the things you can do with just an expression.

Mark44 said:
A small point: this is not an equation. It's important to understand when you're working with an equation and when you're working with an expression, since the kinds of operations you can do are different. For example, if you're working with an equation, you can add a quantity to both sides, multiply both sides by a quantity, and so on, but you're very limited in the things you can do with just an expression.

Ahh, I see!

Thank you very much again! :)

## What is a logarithmic equation?

A logarithmic equation is an equation that involves logarithms. Logarithms are mathematical functions that are used to solve exponential equations. They are the inverse of exponential functions and are written in the form logb(x) = y, where b is the base of the logarithm, x is the argument of the logarithm, and y is the exponent.

## What is the base of a logarithm?

The base of a logarithm is the number that is raised to a certain power to get the argument of the logarithm. In the equation logb(x) = y, b is the base.

## What is the difference between log and ln?

Log and ln are both logarithmic functions, but they have different bases. Log stands for logarithm with base 10, while ln stands for natural logarithm with base e (Euler's number). Logarithms with base 10 are more commonly used in mathematics, while natural logarithms are more commonly used in calculus and other sciences.

## How do you solve a logarithmic equation?

To solve a logarithmic equation, you must use the properties of logarithms to simplify the equation and isolate the variable. The basic properties of logarithms are logb(xy) = logb(x) + logb(y) and logb(x/y) = logb(x) - logb(y). You can also use the change of base formula, logb(x) = log(x)/log(b), to solve for the variable.

## How do you solve the equation [log2 9/log2 10] - log 3?

To solve this equation, we can use the properties of logarithms to rewrite it as log2(9/10) - log2(3). Using the change of base formula, this can be further simplified to [log(9) - log(10)]/log(2) - log(3). Finally, using the property log(x) - log(y) = log(x/y), we can write the equation as log(9/10)/log(2) - log(3). Using a calculator, we can evaluate log(9/10)/log(2) to be approximately -0.152. Therefore, the solution to the equation is -0.152 - log(3).

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