Math  /  Algebra

QuestionCorrect
Use the properties of logarithms to condense the following expression as much as possible, writing the answer as a single term with a coefficient of 1 . All exponents should be positive. ln(x)ln(3z)\ln (x)-\ln (3 z)

Studdy Solution

STEP 1

What is this asking? We need to rewrite the difference of two natural logarithms as a *single* logarithm. Watch out! Remember the logarithm properties, especially the one about subtraction!
Also, make sure the coefficient of the final logarithm is **1**.

STEP 2

1. Apply the logarithm subtraction property.

STEP 3

The logarithm subtraction property states: ln(a)ln(b)=ln(ab). \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right). This property is super useful for simplifying logarithmic expressions!
It tells us that subtracting two logarithms with the same base is the same as taking the logarithm of the quotient of the arguments.
Why does this work?
Think about what logarithms *are*.
They're the inverse of exponentials!
So, if subtracting exponents corresponds to dividing, it makes sense that subtracting logarithms corresponds to dividing their arguments!

STEP 4

In our case, we have ln(x)ln(3z)\ln(x) - \ln(3z).
We can see that a=xa = x and b=3zb = 3z.
Applying the property, we get: ln(x)ln(3z)=ln(x3z). \ln(x) - \ln(3z) = \ln\left(\frac{x}{3z}\right). We've successfully combined the two logarithms into a single logarithm!
Notice how the subtraction outside turned into division inside the logarithm.
So cool!

STEP 5

Our condensed expression is: ln(x3z).\ln\left(\frac{x}{3z}\right).

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