Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 4 - Inverse, Exponential, and Logarithmic Functions

Problem 91

Chapter 5, Problem 91

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b⁴√a)

Verified step by step guidance

Start with the given expression: $\ln \left(b^{4\sqrt{a}}\right)$.

Recall the logarithm power rule: $\ln(x^k) = k \ln(x)$. Apply this to rewrite the expression as $4\sqrt{a} \cdot \ln b$.

Express $\sqrt{a}$ in terms of $a$: $\sqrt{a} = a^{1/2}$. So the expression becomes $4 a^{1/2} \cdot \ln b$.

Since $u = \ln a$ and $v = \ln b$, rewrite $a^{1/2}$ using the exponential and logarithm relationship: $a^{1/2} = e^{(1/2) \ln a} = e^{(1/2) u}$.

Substitute back into the expression to get $4 e^{(1/2) u} \cdot v$, which is the expression in terms of $u$ and $v$ without using the $\ln$ function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the power rule ln(x^r) = r ln(x) and the product rule ln(xy) = ln(x) + ln(y). These allow rewriting complex logarithmic expressions into sums and multiples of simpler logarithms.

Exponent and Root Relationships

Roots can be expressed as fractional exponents, for example, the fourth root of a is a^(1/4). Understanding this allows rewriting expressions like b^4√a as b^4 * a^(1/4), facilitating the use of logarithm properties.

Substitution in Logarithmic Expressions

Given u = ln(a) and v = ln(b), substitution replaces ln(a) and ln(b) with u and v respectively. This helps express logarithmic expressions in terms of u and v without explicitly using the ln function.

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Related Practice

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Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log_√19 5

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Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln √(a³/b⁵)

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Solve each equation. (1/e)^-x= (1/e²)^x+1

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Given that log₁₀ 2 ≈ 0.3010 and log₁₀ 3 ≈ 0.4771, find each logarithm without using a calculator. log₁₀ 6

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Solve each equation for the indicated variable. Use logarithms with the appropriate bases. See Example 10. y = K/(1+ae^-bx), for b

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Textbook Question

Solve each equation for the indicated variable. Use logarithms with the appropriate bases. $I = \frac{E}{R} (1 - e^{- \frac{R t}{2}}), for t$

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b4 √a)

Verified video answer for a similar problem:

Key Concepts

Properties of Logarithms

Exponent and Root Relationships

Substitution in Logarithmic Expressions

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b⁴√a)