QuestionSolve the logarithmic equation. $\log x+\log 4 x=3$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes the sum of the logarithm of $x$ and the logarithm of $4x$ equal to 3. Watch out! Remember the logarithm properties, especially the one about adding logarithms!
Also, be mindful of any restrictions on the values $x$ can take within a logarithm.

STEP 2

1. Combine the logarithms.
2. Rewrite the logarithmic equation as an exponential equation.
3. Solve for $x$ .
4. Check the solution.

STEP 3

We've got $\log x + \log 4x = 3$ .
Remember that adding logarithms is like multiplying the numbers inside!
So, $\log x + \log 4x$ becomes $\log(x \cdot 4x)$ .
This simplifies to $\log(4x^2) = 3$ .
This is awesome because now we have a single logarithm to deal with!

STEP 4

When we see $\log$ without a specified base, it's usually a base-10 logarithm.
So, $\log(4x^2) = 3$ really means $\log_{10}(4x^2) = 3$ .

STEP 5

To get rid of the logarithm, we can rewrite the equation in exponential form.
Remember, logarithms ask "what power do we raise the base to, to get the number inside?" Here, we're saying "10 raised to the power of 3 equals $4x^2$ ".
So, we write $10^3 = 4x^2$ , which is $1000 = 4x^2$ .

STEP 6

We want to get $x$ by itself.
First, let's divide both sides of $1000 = 4x^2$ by 4.
This gives us $\frac{1000}{4} = \frac{4x^2}{4}$ , which simplifies to $250 = x^2$ .

STEP 7

Now, take the square root of both sides to solve for $x$ .
Remember, when we take the square root, we get both a positive and a negative solution.
So, $\sqrt{x^2} = \pm\sqrt{250}$ .
This simplifies to $x = \pm\sqrt{250}$ , which is $x = \pm 5\sqrt{10}$ .

STEP 8

We can't have a negative number inside a logarithm!
Since we have $\log x$ in the original equation, $x$ must be positive.
Therefore, we can't use the negative solution $x = -5\sqrt{10}$ .
So, our only valid solution is $x = 5\sqrt{10}$ .

STEP 9

The solution to the logarithmic equation $\log x + \log 4x = 3$ is $x = 5\sqrt{10}$ .

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