Write each expression in terms of common logarithms, and then give a calculator approximation (correct to four decimal places).
log7^10= 1.183
How do you find the approximation
log3^316 = 5.239
How do you find the approximation
Thank you
log710
=log(10)/log(7)
=1.183294662454939 approximately (use your calculator)
I'll leave the other problem for you to practise.
To find the approximation of logarithms in terms of common logarithms (logarithms with base 10), you can use the following steps:
Step 1: Recognize the given logarithm expression.
In your first example, you are given log7^10. This can be interpreted as "the logarithm of 10 with base 7."
Step 2: Use the change of base formula.
To convert the expression to a common logarithm, you can apply the change of base formula, which states:
log base b of x = log base c of x / log base c of b.
For your first example, we can rewrite log7^10 as log base 10 of 10 / log base 10 of 7.
Step 3: Calculate the values.
Using a calculator or an online logarithm calculator, calculate the values of log base 10 of 10 and log base 10 of 7.
log base 10 of 10 ≈ 1
log base 10 of 7 ≈ 0.8451
Step 4: Calculate the approximation.
Now, divide the value of log base 10 of 10 by the value of log base 10 of 7.
1 / 0.8451 ≈ 1.183
So, the approximation of log7^10 (correct to four decimal places) is approximately 1.183.
For your second example, you are given log3^316. Following the same steps:
Step 1: Recognize the given logarithm expression.
In this case, log3^316 can be understood as "the logarithm of 316 with base 3."
Step 2: Use the change of base formula.
Apply the change of base formula to convert the expression to a common logarithm:
log base 3 of 316 = log base 10 of 316 / log base 10 of 3.
Step 3: Calculate the values.
Calculate the values of log base 10 of 316 and log base 10 of 3 using a calculator or an online logarithm calculator.
log base 10 of 316 ≈ 2.499
log base 10 of 3 ≈ 0.4771
Step 4: Calculate the approximation.
Divide the value of log base 10 of 316 by the value of log base 10 of 3.
2.499 / 0.4771 ≈ 5.2389
So, the approximation of log3^316 (correct to four decimal places) is approximately 5.239.