Questionf) $4 \log _{10} 0.001$

Studdy Solution

STEP 1

What is this asking? We need to calculate the value of a logarithm expression, specifically 4 multiplied by the base-10 logarithm of 0.001. Watch out! Remember the relationship between logarithms and exponents!
Don't mix up the base and the argument.

STEP 2

1. Rewrite the decimal as a fraction
2. Rewrite the fraction as a power of 10
3. Apply the logarithm rule
4. Multiply to find the final result

STEP 3

Let's rewrite our decimal as a fraction!
We've got $0.001$ , which is the same as $\frac{1}{1000}$ .
So, our expression becomes $4 \cdot \log_{10} \frac{1}{1000}$ .
This makes it a bit easier to work with!

STEP 4

Now, let's rewrite that fraction as a power of 10.
We know that $1000 = 10^3$ , so $\frac{1}{1000}$ can be written as $\frac{1}{10^3}$ , which is the same as $10^{-3}$ .
Awesome! Our expression is now $4 \cdot \log_{10} 10^{-3}$ .
This is starting to look much simpler!

STEP 5

Time to use that logarithm power rule!
Remember, $\log_a (x^b) = b \cdot \log_a x$ .
In our case, $x = 10$ and $b = -3$ .
So, $\log_{10} 10^{-3}$ becomes $-3 \cdot \log_{10} 10$ .

STEP 6

And since $\log_a a = 1$ (because $a^1 = a$ ), we know that $\log_{10} 10 = 1$ .
So, our expression simplifies to $4 \cdot (-3 \cdot 1)$ .

STEP 7

Let's multiply those numbers together! $4 \cdot (-3 \cdot 1) = 4 \cdot -3 = -12$ .
And there we have it!

STEP 8

$4 \log_{10} 0.001 = -12$ .

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