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Chapter 4: Problem 41

Evaluate each expression without using a calculator. $$8^{\log _{8} 19}$$

Short Answer

Expert verified
The value of the expression \(8^{\log _{8} 19}\) is \(19\).

Step by step solution

01

Identify the Main Logarithmic Rule

The main rule used in this exercise is \(a^{\log_a(b)} = b\). This is one of the basic rules of logarithms, stating that if a certain number \(a\) is raised to a logarithm of the same base \(a\), then the results equals the number within the logarithm.
02

Apply the Logarithmic Rule

Applying this rule to our expression \(8^{\log _{8} 19}\), the base of the exponent and the base of the logarithm are the same, being \(8\). Hence, we can directly apply the logarithmic rule to simplify this expression, making it equivalent to the number inside the logarithm, which is \(19\).

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Most popular questions from this chapter

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(=2.2\) pounds) Use a graphing utility to graph the function. Then \([\mathrm{TRACE}]\) along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Describe the relationship between an equation in logarithmic form and an equivalent equation in exponential form.
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