Write the expression as a single logarithm.

log_3 40-log_3 10

I'm completely confused with logarithmic equations!! Can someone please please help me understand?

since loga - logb = log(a/b), we have

log_3 40-log_3 10
= log_3(40/10)
= log_3(4)

logs are just exponents in reverse. You know that

x^6/x^4 = x^(6-4)

log_3(40) is the power of 3 you need to get 40

since ^ and log are inverse operations,

3^(log_3(x)) = x
log_3(3^x) = x

just as
√x^2 = (√x)^2 = x
(x/2)*2 = (x*2)/2 = x
(x+2)-2 = (x-2)+2 = x

Sure, I'd be happy to help you understand logarithmic equations. But first, let's have a little fun with this problem.

Why did the logarithm look for a new job? Because it felt "loggy" at work!

Now, let's get serious and solve the problem. In order to combine the two logarithms into a single logarithm, we can use the properties of logarithms. One of these properties states that when subtracting logarithms with the same base, we can rewrite it as the logarithm of the division of the two numbers inside the logarithm.

So, let's apply this property to our problem. We have:

log_3 40 - log_3 10

Using the property, we can rewrite this expression as:

log_3 (40/10)

Simplifying further, we have:

log_3 4

And there you have it! The expression log_3 40 - log_3 10 can be written as a single logarithm log_3 4.

Now, don't get too "loggy" with logarithmic equations! If you have any more questions, feel free to ask.

Of course! I'll break it down step-by-step to help you understand logarithmic equations.

Step 1: Recall the properties of logarithms.
The properties of logarithms can be used to simplify and manipulate logarithmic expressions. One of the key properties is the subtraction property:

log_b(x) - log_b(y) = log_b(x/y)

This property allows us to combine two logarithms with the same base by subtracting them.

Step 2: Apply the subtraction property.
Using the subtraction property, we can rewrite the expression:

log_3 40 - log_3 10 = log_3 (40/10)

Step 3: Simplify the expression.
In this case, 40/10 simplifies to 4. Therefore, the expression can be further simplified as:

log_3 4

Step 4: Write the expression as a single logarithm.
The final answer is log_3 4. This represents the logarithm with base 3 of the number 4.

I hope this helps clarify logarithmic equations for you! Let me know if you have any other questions.

Sure, I'd be happy to help you understand logarithmic equations!

To simplify the expression `log_3 40 - log_3 10`, we can use a property of logarithms called the quotient rule. The quotient rule states that the difference of two logarithms with the same base is equal to the logarithm of the quotient of the numbers inside the logarithms. In other words:

log_a (b) - log_a (c) = log_a (b/c)

Using this rule, we can simplify the expression as follows:

log_3 40 - log_3 10 = log_3 (40/10)

Now, simplify the expression inside the logarithm:

= log_3 (4)

Therefore, the expression `log_3 40 - log_3 10` simplifies to `log_3 4`.