Verified Answer :

  • Apply the power rule \(n \ln(a) = \ln(a^n)\) to shift coefficients into exponents: \(4 \ln(2) = \ln(16)\), \(5 \ln(x) = \ln(x^5)\), and \(\frac{1}{2} \ln(x + 3) = \ln(\sqrt{x + 3})\).
  • Combine the first two terms using the product rule \(\ln(a) + \ln(b) = \ln(ab)\) to get \(\ln(16x^5)\).
  • Subtract the third term using the quotient rule \(\ln(a) - \ln(b) = \ln(a/b)\) to get the final single logarithmic form \(\ln\left(\frac{16x^5}{\sqrt{x+3}}\right)\).
  • The final simplified expression is \(\boxed{\ln\left(\frac{16x^5}{\sqrt{x+3}}\right)}\).

Explanation

  1. Analysis of Logarithmic Properties

The goal is to combine the logarithmic expression \(4 \ln(2) + 5 \ln(x) - \frac{1}{2} \ln(x + 3)\) into a single logarithmic term. We recall the relevant Laws of Logarithms:

  1. The power rule: \(n \ln(a) = \ln(a^n)\).
  2. The product rule: \(\ln(a) + \ln(b) = \ln(ab)\).
  3. The quotient rule: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).

These laws will allow us to condense the sum and difference of the individual logarithmic terms into a single, compact expression.

  1. Applying the Power Rule

First, we apply the power rule, \(n \ln(a) = \ln(a^n)\), to move the coefficients into the arguments of each logarithm:

  • For the first term: \(4 \ln(2) = \ln(2^4) = \ln(16)\).
  • For the second term: \(5 \ln(x) = \ln(x^5)\).
  • For the third term: \(\frac{1}{2} \ln(x + 3) = \ln((x + 3)^{1/2}) = \ln(\sqrt{x + 3})\).

After applying these transformations, our expression becomes:
\[\ln(16) + \ln(x^5) - \ln(\sqrt{x + 3})\]

  1. Applying the Product Rule

Next, we use the product rule, \(\ln(a) + \ln(b) = \ln(ab)\), to combine the first two terms:
\[\ln(16) + \ln(x^5) = \ln(16 \cdot x^5) = \ln(16x^5)\]
The expression now is:
\[\ln(16x^5) - \ln(\sqrt{x + 3})\]

  1. Applying the Quotient Rule and Conclusion

Finally, we use the quotient rule, \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), to combine the remaining terms:
\[\ln(16x^5) - \ln(\sqrt{x + 3}) = \ln\left(\frac{16x^5}{\sqrt{x + 3}}\right)\]
This is our combined, single logarithmic expression. The final result is \(\boxed{\ln\left(\frac{16x^5}{\sqrt{x+3}}\right)}\).

Examples

Logarithmic functions are used in many scientific fields, such as chemistry to measure acidity (pH scale) or in acoustics to measure sound intensity (decibels). Combining multiple logarithmic terms into a single expression allows scientists to simplify complex formulas, making them easier to calculate or differentiate. For example, if you are analyzing the relationship between the concentration of different reactants in a chemical reaction over time, you might encounter a sum of logarithms that can be simplified into a single, more manageable term representing the overall reaction rate ratio.

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