# How do you solve #log_10(4x)-log_10(12+sqrtx)=2#?

Thus, using the log definition, we have

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To solve the equation (\log_{10}(4x) - \log_{10}(12+\sqrt{x}) = 2), we can use the properties of logarithms to simplify the equation and then solve for (x).

First, apply the quotient rule of logarithms, which states that (\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)). So, we rewrite the equation as a single logarithm:

[\log_{10}\left(\frac{4x}{12+\sqrt{x}}\right) = 2]

Next, use the definition of logarithms to rewrite the equation in exponential form:

[10^2 = \frac{4x}{12+\sqrt{x}}]

Simplify by multiplying both sides by (12+\sqrt{x}):

[100(12+\sqrt{x}) = 4x]

Expand:

[1200 + 100\sqrt{x} = 4x]

Rearrange the equation into a quadratic form by bringing all terms to one side:

[4x - 100\sqrt{x} - 1200 = 0]

Now, let (u = \sqrt{x}), so (u^2 = x). Substituting, the equation becomes:

[4u^2 - 100u - 1200 = 0]

This is now a quadratic equation in (u). Solve for (u) using the quadratic formula:

[u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}]

where (a = 4), (b = -100), and (c = -1200).

After finding the values of (u), substitute them back into (x = u^2).

Check for extraneous solutions, as the original equation involves a logarithm, and logarithms are only defined for positive values.

Finally, verify your solutions by substituting them back into the original equation to ensure they satisfy the equation.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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