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Chapter 4: Problem 17

Simplify the following expressions. $$c^{\ln x+\ln 2}$$

Short Answer

Expert verified
(2x)^{c}

Step by step solution

01

Use Logarithm Property on Exponents

The expression given is simplified using the property of logarithms that states \(\text{ln}(a) + \text{ln}(b) = \text{ln}(a \times b)\). Therefore, the exponent can be rewritten as \(\text{ln}(x) + \text{ln}(2) = \text{ln}(2x)\).
02

Simplify using the Exponential Property

The next step is to use the property of exponents that \(a^{\text{ln}(b)} = b^{\text{ln}(a)}\). Therefore, \(c^{\text{ln}(2x)}\) simplifies to \(e^{\text{ln}(c^{\text{ln}(2x)})} = (2x)^{\text{ln}(c)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithmic Addition
Logarithmic addition is a fundamental property of logarithms. It allows us to combine logarithms of products into a single logarithm. The property states that \(\text{ln}(a) + \text{ln}(b) = \text{ln}(a \times b)\). This is useful in simplifying expressions where multiple logarithms are present. For instance, in the expression \(\text{ln}(x) + \text{ln}(2)\), we can combine the logs into one: \( \text{ln}(x) + \text{ln}(2) = \text{ln}(2x)\). This simplification makes it easier to manage and manipulate the expression in later steps.
Exponential Properties
Exponential properties are rules that apply to exponents and logarithms, helping to simplify complex expressions. One key property is: if you have an exponent in the form \(a^{\text{ln}(b)}\), it can be rewritten as \(b^{\text{ln}(a)}\). This rearrangement stems from the inverse relationship between logarithms and exponents. For example, consider the expression \(c^{ \text{ln}(2x)}\). Using the property mentioned, this can be transformed to \(e^{\text{ln}(c^{\text{ln}(2x)})} = (2x)^{\text{ln}(c)}\). These properties are instrumental in simplifying exponential functions and making them easier to work with.
Simplifying Expressions
Simplifying expressions is a crucial part of solving mathematical problems. It involves reducing a complex expression to its simplest form. In our example, we start by simplifying the exponent through logarithmic addition: \( \text{ln}(x) + \text{ln}(2) = \text{ln}(2x)\). Then we utilize the exponential property to further simplify: \(c^{\text{ln}(2x)} = e^{\text{ln}(c^{\text{ln}(2x)})} = (2x)^{\text{ln}(c)}\). Throughout the simplification process, we employ various mathematical properties to make the expression as straightforward as possible. By understanding and applying these fundamental rules, we can confidently tackle and simplify a wide range of mathematical expressions.

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Most popular questions from this chapter

The highest price ever paid for an artwork at auction was for Pablo Picasso's 1955 painting Les femmes d'Alger, which fetched \(\$ 179.4\) million in a Christie's auction in \(2015 .\) The painting was last sold in 1997 for \(\$ 31.9\) million. If the painting keeps on appreciating at its current rate, then a model for its value is given by \(f(t)=31.87 e^{0.096 t},\) where \(f(t)\) is in millions of dollars and \(t\) is the number of years since 1997

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