Math Snap

STEP 1

What is this asking?
Rewrite this logarithmic equation in exponential form.
Watch out!
Remember the relationship between logs and exponents: $\log_b(a) = c$ means $b^c = a$.
Don't mix up the base, exponent, and result!

STEP 2

1. Identify the components of the logarithmic equation.
2. Rewrite the equation in exponential form.

STEP 3

Alright, let's break down this logarithmic equation!
We've got $\log \left(\frac{1}{\sqrt[3]{10}}\right)=-\frac{1}{3}$.
Remember, when the base isn't explicitly written, it's understood to be 10.
So, we can rewrite it as $\log_{10} \left(\frac{1}{\sqrt[3]{10}}\right)=-\frac{1}{3}$.

STEP 4

Now, let's identify the key players: the base is $10$, the result is $\frac{1}{\sqrt[3]{10}}$, and the exponent is $-\frac{1}{3}$.
It's like a puzzle, and we've got all the pieces!

STEP 5

Remember the fundamental relationship between logs and exponents: $\log_b(a) = c$ means $b^c = a$.
Let's apply this to our equation.

STEP 6

Our base is $10$, so that's the base of our exponential expression.
Our exponent is $-\frac{1}{3}$, so that's what we raise $10$ to.
And our result is $\frac{1}{\sqrt[3]{10}}$.

STEP 7

Putting it all together, we get $10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}}$.
Boom! We've successfully rewritten the logarithmic equation as an exponential equation.

The equivalent exponential equation is $10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}}$.