Given 2 =10^a and 9 =10^b
Determine the values of the following:
1. Log4
2. Log 6
Given 2 =10^a and 9 =10^b, you know that
a = log2
b = log9
Now note that
log4 = log(2^2) = 2log2
log6 = log2 + log3 = log2 + 1/2 log9
...
1. Log4? Oh, that's a tricky one! Well, if we rewrite 4 as 10^c, then we just need to solve for c. Let's see, since 2 = 10^a, that means a = log2. And since 9 = 10^b, that means b = log9. So, we need to find c in 10^c = 4. Hmmm... Maybe we can ask a mathematician to solve this one for us!
2. Log6? Well, let's put our thinking cap on and give it a shot! If we rewrite 6 as 10^d, then we just need to solve for d. From the given equations, we know a = log2 and b = log9. So, we need to find d in 10^d = 6. Now, I could start spouting logarithmic formulas, but why not just ask Siri or Google? They seem like the type who know their logarithms!
To determine the values of logarithms, we need to use the properties of logarithms and the given equations.
1. Log4:
We can rewrite the equation 2 = 10^a as a = log10(2). Using the property logb(b^x) = x, we have a = log10(2) = log10(10^a) = a log10(10). Therefore, a = 1.
Now, we can find log4 using the change of base formula: log_b(x) = log_c(x) / log_c(b).
log4 = log10(4) / log10(2)
Since 4 = 2^2, log4 can be written as:
log4 = 2 log10(2) / log10(2)
Simplifying the expression, we get:
log4 = 2
2. Log 6:
Similarly, we can rewrite the equation 9 = 10^b as b = log10(9). Using the property logb(b^x) = x, we have b = log10(9) = log10(10^b) = b log10(10). Therefore, b = 1.
Now, we can find log6 using the change of base formula: log_b(x) = log_c(x) / log_c(b).
log6 = log10(6) / log10(2)
Since 6 = 2 * 3, log6 can be written as:
log6 = log10(2 * 3) / log10(2)
Using the property log_b(m * n) = log_b(m) + log_b(n), we can simplify the expression as follows:
log6 = (log10(2) + log10(3)) / log10(2)
That is the simplified expression for log6.
To find the values of log4 and log6, we can use the properties of logarithms and solve the equations provided.
1. To determine the value of log4, we need to find the power to which 10 must be raised to equal 4. Let's rewrite the equation 2 = 10^a in logarithmic form:
10^a = 2
Taking the logarithm (base 10) of both sides, we get:
log(10^a) = log(2)
Using the power rule of logarithms, we can bring the exponent down to the front:
a * log(10) = log(2)
Since log(10) equals 1, we can simplify the equation to:
a = log(2)
Therefore, the value of log4 is log(2).
2. Similarly, to find the value of log6, we rewrite the equation 9 = 10^b in logarithmic form:
10^b = 9
Taking the logarithm (base 10) of both sides:
log(10^b) = log(9)
Using the power rule of logarithms:
b * log(10) = log(9)
Again, as log(10) equals 1:
b = log(9)
So, the value of log6 is log(9).