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Chapter 3: Problem 19

Evaluate the indicated expression. Do not use a calculator for these exercises. $$ \log \sqrt{1000} $$

Short Answer

Expert verified
The value of the expression \(\log \sqrt{1000}\) is \(\frac{3}{2}\).

Step by step solution

01

Rewrite the square root

We can rewrite the square root of 1000 as an exponential expression in the form of 1000 raised to the power of 1/2. So, the expression becomes: $$ \log(1000^{\frac{1}{2}}) $$
02

Use the power rule of logarithms

The power rule for logarithms states that for any base b, $$ \log_b(a^n) = n\log_b(a) $$ In this case, the base is 10 (since there's no base written, we assume it's the common logarithm, which has a base of 10). So we can apply this rule to our expression: $$ \frac{1}{2}\log(1000) $$
03

Evaluate the logarithm

Now we will evaluate the logarithm of 1000. Remember that the logarithm of a number x with base 10 is the power to which 10 must be raised to obtain the number x. In other words, we need to find the exponent n when \(10^n = 1000\). Since \(10^3 = 1000\), the logarithm of 1000 is 3. Replacing the logarithm with its value, we get: $$ \frac{1}{2} \times 3 $$
04

Multiply the values

Finally, we can multiply 1/2 and 3 to get the value of the expression: $$ \frac{1}{2} \times 3 = \frac{3}{2} $$ So, the value of the expression \(\log \sqrt{1000}\) is \(\frac{3}{2}\).

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Most popular questions from this chapter

Show that for every positive number \(c,\) we have $$ \ln (c+t)-\ln c \approx \frac{t}{c} $$ for small values of \(t\).

Show that the range of cosh is the interval \([1, \infty)\).

Estimate the indicated value without using a calculator. \(\frac{e^{9}}{e^{8.997}}\)

Estimate the given number. Your calculator will be unable to evaluate directly the expressions in these exercises. Thus you will need to do more than button pushing for these exercises. \(\left(1+10^{-100}\right)^{3 \cdot 10^{100}}\)

Show that the range of \(\sinh\) is the set of real numbers.
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