Given that log3=0.4771 and iog5=0.6990 evaluate the value of log1.35+log2.25
log1.35+log2.25
= log(135 * 10^-2) + log(225 * 10^-2)
= log135 - 2 + log225 - 2
= log(5*27) + log(9*25) - 4
= log5 + 3log3 + 2log3 - 2log5 - 4
= 5log3 - log5 - 4
...
given that log 3=0.4771 and log 5 =0.699 find long 135 without using calcukator or tables
To evaluate the value of log1.35 + log2.25, we can use the properties of logarithms.
First, let's simplify the equation:
log1.35 + log2.25
Using the property of addition of logarithms, we can combine these logarithms into a single logarithm:
log1.35 * 2.25
Next, let's simplify further:
log(1.35 * 2.25)
Now, evaluate the expression within the logarithm:
log(3.0375)
Since we are given that log3 = 0.4771, we can use this information to evaluate the expression further:
log(3.0375) = log(3 * 1.0125) = log(3) + log(1.0125) = 0.4771 + log(1.0125)
Now, we need to find the value of log(1.0125). However, we are only given log3 and log5. We don't have log2. So, we need to find a way to express 1.0125 in terms of 3 and 5.
To do this, we need to find the logarithm base that both 1.0125 and 3 have in common. In this case, the closest option would be log5, since 5 is closer to 1.0125 than 3.
So, let's express 1.0125 in terms of 5:
1.0125 = 5^x
To solve for x, we can take the logarithm of both sides with the same base, which is log5 in this case:
log5(1.0125) = x
Using the given value, log5 = 0.6990:
0.6990 ≈ x
So, we can approximate log(1.0125) as 0.6990.
Therefore, log1.35 + log2.25 is equal to approximately 0.4771 + 0.6990 = 1.1761.
To evaluate the expression log1.35 + log2.25, we can use some basic logarithmic properties. One of these properties states that the sum of the logarithms of two numbers is equal to the logarithm of their product.
So, we can rewrite the expression as log(1.35 * 2.25).
To find the value of log(1.35 * 2.25), we need to multiply 1.35 and 2.25 and then find the logarithm of the result.
Using a calculator, we find that 1.35 * 2.25 is equal to 3.0375.
Now, we need to find the logarithm of 3.0375. However, we do not have the logarithm of 3.0375 directly given in the question.
But we do have the logarithms of 3 and 5 given: log3 = 0.4771 and log5 = 0.6990.
We can use these known values to approximate the logarithm of 3.0375 using interpolation.
Interpolation involves finding the value between two known values by proportionally estimating the unknown value.
To approximate log3.0375, we can first find the proportion of log3.0375 with respect to log3 and log5.
The proportion of log3.0375 with respect to log3 is (3.0375/3) ≈ 1.0125.
Similarly, the proportion of log3.0375 with respect to log5 is (3.0375/5) ≈ 0.6075.
Now, we can use these proportions to approximate the value of log3.0375:
log3.0375 ≈ log3 + (log3 - log5) * 1.0125.
Substituting the given values, we have:
log3.0375 ≈ 0.4771 + (0.4771 - 0.6990) * 1.0125.
Evaluating the expression, we find:
log3.0375 ≈ 0.4771 + (-0.2219) * 1.0125
log3.0375 ≈ 0.4771 - 0.2242
log3.0375 ≈ 0.2529
Therefore, log1.35 + log2.25 is approximately equal to 0.2529.