Question

question 39 of 40
evaluate the common logarithm without the use of a calculator.
log0.000001
log0.000001 =
question 40 of 40
write the equation in logarithmic form.
2^5 = 32
2^5 = 32 written in logarithmic form is .

Explanation:

Question 39

Step1: Recall the definition of common logarithm

The common logarithm \( \log x \) is base - 10, i.e., \( \log_{10}x \). We know that if \( y = \log_{10}x \), then \( x = 10^{y} \). We need to find \( y \) such that \( 10^{y}=0.000001 \).

Step2: Express 0.000001 as a power of 10

We know that \( 0.000001=\frac{1}{1000000}=\frac{1}{10^{6}} = 10^{- 6} \).
Since \( \log_{10}(10^{y})=y \) (by the property of logarithms \( \log_{a}(a^{x}) = x \) for \( a>0,a
eq1 \)), and \( 0.000001 = 10^{-6} \), then \( \log(0.000001)=\log_{10}(10^{-6})=-6 \).

Step1: Recall the definition of logarithmic form

For an exponential equation \( a^{b}=c \) (where \( a > 0,a
eq1 \)), the corresponding logarithmic form is \( \log_{a}c = b \).

Step2: Identify \( a \), \( b \), and \( c \) from the given equation

In the equation \( 2^{5}=32 \), we have \( a = 2 \), \( b = 5 \), and \( c = 32 \).
Using the conversion formula from exponential to logarithmic form, we get \( \log_{2}32 = 5 \).

Answer:

\( \log0.000001=-6 \)

Question 40