If 8.2^p = 10.5^q, then p/q = ?
Estimate your answer to two decimal places.
p log(8.2) = q log(10.5)
p / q = log(10.5) / log(8.2)
To solve for p/q, we can take the logarithm of both sides of the equation 8.2^p = 10.5^q.
Step 1: Take the logarithm of both sides using a common logarithm (log base 10) or a natural logarithm (ln, log base e). Let's use the natural logarithm (ln) for this example.
ln(8.2^p) = ln(10.5^q)
Step 2: Use the logarithm properties to simplify the equation. According to the logarithm rules, ln(a^b) = b * ln(a).
p * ln(8.2) = q * ln(10.5)
Step 3: Divide both sides of the equation by q * ln(8.2) to isolate p/q.
p/q = ln(10.5) / ln(8.2)
Using a calculator, we can approximate the values of the natural logarithms:
ln(10.5) ≈ 2.351
ln(8.2) ≈ 2.104
Therefore, p/q ≈ 2.351 / 2.104 ≈ 1.12
To solve for p/q in the equation 8.2^p = 10.5^q, we can take the logarithm of both sides. The most commonly used logarithm is the natural logarithm (ln) which has the base e.
So, taking the natural logarithm of both sides, we have:
ln(8.2^p) = ln(10.5^q)
Using the logarithmic property that ln(a^b) = b * ln(a), we can simplify to:
p * ln(8.2) = q * ln(10.5)
To find p/q, we need to isolate it. We can divide both sides by q * ln(10.5):
p/q = ln(10.5) / ln(8.2)
Using a calculator, we can evaluate ln(10.5) ≈ 2.3514 and ln(8.2) ≈ 2.1041.
Substituting these values into the equation, we get:
p/q ≈ 2.3514 / 2.1041 ≈ 1.1165
Therefore, p/q is approximately 1.12 when rounded to two decimal places.