Question

Write each expression as a sum and/or difference of logarithms. Express powers as factors. $$ \ln \left(x \sqrt{1+x^{2}}\right) \quad x>0 $$

Short answer

\[ \ln(x) + \frac{1}{2} \ln(1+x^2) \]

Step-by-step solution

Apply the logarithm product rule

Recall the logarithm product rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \) \In this exercise, let \( a = x \) and \( b = \sqrt{1+x^2} \). Therefore, rewrite the logarithm as: \( \ln(x \sqrt{1+x^2}) = \ln(x) + \ln(\sqrt{1+x^2}) \)

Apply the logarithm power rule

Recall the logarithm power rule: \( \ln(a^b) = b \ln(a) \) \In this case, \( \sqrt{1+x^2} = (1+x^2)^{1/2} \). Thus, \ln(\sqrt{1+x^2}) \) can be expressed as \( \( \frac{1}{2} \ln(1+x^2) \). So, rewrite the second term accordingly: \ \ln(x) + \ln(\sqrt{1+x^2}) = \ln(x) + \frac{1}{2} \ln(1+x^2)}

Key concepts

product rule

In logarithms, the product rule is a fundamental property. It states that the logarithm of a product is equal to the sum of the logarithms of its factors.
This can be written in mathematical notation as: \( \ln(a \cdot b) = \ln(a) + \ln(b) \).
This rule helps in breaking down complex logarithmic expressions into simpler parts.

In the given exercise, we used the product rule on the expression \( \ln(x \sqrt{1+x^{2}}) \).
We considered \( a = x \) and \( b = \sqrt{1+x^{2}} \).
By applying the product rule, we separated the logarithm into: \( \ln(x) + \ln(\(\sqrt{1+x^{2}}\)) \).
This decomposition made it easier to further simplify the expression.

power rule

The power rule for logarithms states that the logarithm of a number raised to a power is the power times the logarithm of the number.
This can be written as: \( \ln(a^{b}) = b \ln(a) \).

In our problem, we needed to handle the term \( \ln(\(\sqrt{1+x^{2}}\)) \).
We first recognized that \( \sqrt{1+x^{2}} \) can be written as \( (1+x^2)^{1/2} \).
Using the power rule, we could then express this as \( \ln(\(1+x^2\)^{1/2}) = \frac{1}{2} \ln(1+x^2) \).
By applying the power rule, the logarithm involving a square root was converted into a simpler form involving a multiplication.

sum and difference of logarithms

Writing logarithmic expressions as a sum or difference of terms can make calculations easier.
The product rule and quotient rule are key tools in this process:

• Product Rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
• Quotient Rule: \( \ln(\(\frac{a}{b}\)) = \ln(a) - \ln(b) \)

These rules facilitate breaking down complex logarithmic relationships into simpler parts.

For the given problem, the final step includes combining the results of applying the product and power rules.
We simplified the expression \( \ln(x \sqrt{1+x^2}) \) into two simpler logarithmic terms: \( \ln(x) \) and \( \frac{1}{2} \ln(1+x^2) \).

Therefore, the original logarithmic expression can be written as: \( \ln(x) + \frac{1}{2} \ln(1+x^2) \).
This new form, expressed as a sum of logarithms, is easier to handle and understand.