Use the change-of-base formula to evaluate each logarithm correct to six decimal places.
a. log base 3 of 18
b. log base 18 of 9
a. To evaluate log base 3 of 18 using the change-of-base formula, we will use log base 10 and log base 3.
First, we can write the change-of-base formula as:
log base a of x = log base b of x / log base b of a
In this case, a=3 and x=18, so we can rewrite the logarithm as:
log base 3 of 18 = log base 10 of 18 / log base 10 of 3
Now, we can evaluate the logarithms using a scientific calculator:
log base 10 of 18 ≈ 1.255273
log base 10 of 3 ≈ 0.477121
Plugging these values back into the formula:
log base 3 of 18 ≈ 1.255273 / 0.477121 ≈ 2.630929
Therefore, log base 3 of 18 is approximately 2.630929, correct to six decimal places.
b. To evaluate log base 18 of 9 using the change-of-base formula, we will use log base 10 and log base 18.
Using the same change-of-base formula as before:
log base a of x = log base b of x / log base b of a
In this case, a=18 and x=9, so we can rewrite the logarithm as:
log base 18 of 9 = log base 10 of 9 / log base 10 of 18
Using a scientific calculator again:
log base 10 of 9 ≈ 0.954243
log base 10 of 18 ≈ 1.255273
Plugging these values back into the formula:
log base 18 of 9 ≈ 0.954243 / 1.255273 ≈ 0.759836
Therefore, log base 18 of 9 is approximately 0.759836, correct to six decimal places.
a. To evaluate log base 3 of 18, we can use the change-of-base formula which states that for any positive real numbers x and b, where b ≠ 1:
log base b of x = log base a of x / log base a of b
We can choose any base a for which we know the logarithms. Common choices are base 10 (log base 10) and base e (natural logarithm, log base e). Let's choose base 10.
Using the change-of-base formula, we have:
log base 3 of 18 = log base 10 of 18 / log base 10 of 3
Now, let's calculate each part of the equation.
First, let's find log base 10 of 18 using a calculator:
log base 10 of 18 ≈ 1.255273
Next, let's find log base 10 of 3 using a calculator:
log base 10 of 3 ≈ 0.477121
Now, let's substitute these values back into the change-of-base formula:
log base 3 of 18 ≈ 1.255273 / 0.477121
Calculating this division gives:
log base 3 of 18 ≈ 2.629154
Therefore, log base 3 of 18 is approximately 2.629154, correct to six decimal places.
b. Similarly, to evaluate log base 18 of 9, we can use the change-of-base formula with base 10:
log base 18 of 9 = log base 10 of 9 / log base 10 of 18
Let's calculate each part of the equation.
First, log base 10 of 9:
log base 10 of 9 ≈ 0.954243
Next, log base 10 of 18:
log base 10 of 18 ≈ 1.255273
Now, substitute these values back into the change-of-base formula:
log base 18 of 9 ≈ 0.954243 / 1.255273
Calculating this division gives:
log base 18 of 9 ≈ 0.760899
Therefore, log base 18 of 9 is approximately 0.760899, correct to six decimal places.
To evaluate logarithms using the change-of-base formula, we need to convert the given logarithm to a logarithm with a base that is convenient to evaluate. The change-of-base formula states that for any positive real numbers a, b, and c, where c > 0 and not equal to 1:
log base c of a = log base b of a / log base b of c
Now let's apply the change-of-base formula to evaluate the given logarithms:
a. log base 3 of 18:
Using the change-of-base formula, we can rewrite this logarithm with a more convenient base, such as 10 or the natural logarithm base e. Let's choose base 10 for this example:
log base 3 of 18 = log base 10 of 18 / log base 10 of 3
To evaluate this using a calculator:
1. Calculate log base 10 of 18.
2. Calculate log base 10 of 3.
3. Divide the result of step 1 by the result of step 2.
b. log base 18 of 9:
Similar to the previous example, we'll use the change-of-base formula to rewrite this logarithm:
log base 18 of 9 = log base 10 of 9 / log base 10 of 18
Again, we'll use base 10 for convenience.
To evaluate this using a calculator:
1. Calculate log base 10 of 9.
2. Calculate log base 10 of 18.
3. Divide the result of step 1 by the result of step 2.
Remember to round the final answer to six decimal places as requested.