Can someone explain how to figure out these types of problems?
Solve log3 (x - 4) > 2.
Evaluate log7 49.
If log5 4 ≈ .8614 and log5 9 ≈ 1.3652 find the approximate value of log5 36.
Solve the equation, log4 (m + 2) - log4 (m -5) = log4 8.
Solve, 2^x = 15.
You must know the meaning of logs
Memorize this equivalence and the following example:
loga x = y <-----> a^u = x
log2 8 = 3 <-----> 2^3 = 8
so consider log3 (x - 4) = 2
---> 3^2 = x-4
13 = x
so clearly if log3 (x - 4) > 2
x-4 > 3^2
x > 13
check: let x = 31
log3(31-4) > 2
log3 27 > 2
is 3^3 > 2 ? YES
using the definition of logs I just gave you, you should be able to evaluate log7 49
log5 36 = log5 (9 * 4) = log5 9 + log5 4
you are given both of them, so just add the given logs
log4 (m + 2) - log4 (m -5) = log4 8
log4((m+2)/(m-5) = log4 8
so clearly:
(m+2)/(m-5) = 8
8m - 40 = m+2
7m = 42
m = 6
2^x = 15
take log10 of both sides
then
log 2^x = log 15
x log2 = log15
x = log15/log2
= .... , use your calculator
Sure! I can help you figure out these types of problems.
1. Solve log3(x - 4) > 2:
In order to solve this logarithmic inequality, we need to isolate the logarithm on one side of the inequality symbol. Start by exponentiating both sides of the inequality using the base of the logarithm, which in this case is 3. This gives us:
3^(log3(x - 4)) > 3^2
x - 4 > 9
Now, solve for x by adding 4 to both sides of the inequality:
x > 13. Since we are dealing with a strict inequality (">"), the solution is x > 13.
2. Evaluate log7(49):
To evaluate this logarithmic expression, we need to determine the exponent that 7 needs to be raised to in order to obtain 49. In this case, 7^2 = 49. Therefore, log7(49) = 2.
3. Find the approximate value of log5(36):
To find the approximate value of log5(36), we can use the properties of logarithms. Since log5(36) is not a known value, we can express it in terms of the given logarithmic values. We know that 36 can be written as 6^2, so we have:
log5(6^2) = 2 * log5(6)
Now, we can substitute the given logarithmic values:
2 * log5(6) ≈ 2 * (log5(4) + log5(9))
≈ 2 * (0.8614 + 1.3652)
≈ 2 * 2.2266
≈ 4.4532. Therefore, the approximate value of log5(36) is 4.4532.
4. Solve the equation log4(m + 2) - log4(m - 5) = log4(8):
To solve this equation involving logarithms, we can use the properties of logarithms. Start by combining the logarithms on the left side of the equation:
log4((m + 2)/(m - 5)) = log4(8).
Since both sides of the equation have the same base (log4), we can equate their arguments:
(m + 2)/(m - 5) = 8.
Now, solve for m by cross-multiplying and simplifying the equation:
m + 2 = 8(m - 5)
m + 2 = 8m - 40
40 - 2 = 8m - m
38 = 7m
m = 38/7. Therefore, the solution to the equation is m = 38/7.
5. Solve the equation 2^x = 15:
To solve an exponential equation like this, we need to use logarithms. Take the logarithm of both sides using any base you prefer (common bases such as 10 or e are commonly used):
log(2^x) = log(15).
Using the logarithmic property: log(a^b) = b*log(a), we can rewrite the equation as:
x * log(2) = log(15).
Now, divide both sides of the equation by log(2):
x = log(15) / log(2).
Use a calculator to evaluate log(15) and log(2), then perform the division to find the approximate value of x.