Given that log5 3=0.683 and log5 7=1.209.without using calculator,evaluate
(a) log5 1.4
b)log7 75
c) log3 125
Did you notice that 1.4 = 7/5 ?
log5 1.4
= log5 (7/5)
= log5 7 - log5 5
= 1.209 - 1
= .209
A property of logs that comes in handy here is
loga b = 1/logb a
b) log7 75
= log7 (3*5^2)
=log7 3 + 2log7 5
= (log5 3) / (log5 7) + 2/log5 7)
= .683/1.209 + 2/1.209
= 2.683/1.209
= 2683/1209 ----> At this point I used Calculator to get appr 2.219
c) log3 125
= log3 5^3
= 3 log3 5
= 3 (1/log5 3)
= 3/.683
To evaluate the given logarithmic expressions without using a calculator, we can use the properties of logarithms.
(a) To find log5 1.4:
We know that log5 3 = 0.683 and log5 7 = 1.209.
We can express 1.4 as a combination of 3 and 7:
1.4 = 3 * (1.4/3) = 3 * (0.467)
Using the property log ab = log a + log b:
log5 1.4 = log5 (3 * 0.467) = log5 3 + log5 0.467
Now, to find log5 0.467:
We know that log5 1 = 0 (since 5 raised to the power of 0 equals 1).
We can express 0.467 as a power of 5:
0.467 ≈ 5^x, where x is the value we are trying to find.
Taking logarithms of both sides:
log5 0.467 = log5 5^x
Using the property log a^b = b * log a:
log5 0.467 = x * log5 5
Since log5 5 = 1, we can simplify:
log5 0.467 = x * 1 = x
Therefore, x ≈ log5 0.467
Now, we can calculate log5 1.4:
log5 1.4 = log5 3 + log5 0.467 ≈ 0.683 + log5 0.467
Now, approximate the value using the given logarithmic values:
log5 1.4 ≈ 0.683 + 0.467 ≈ 1.15
So, log5 1.4 ≈ 1.15.
(b) To find log7 75:
We need to express 75 as a combination of 7 and other numbers.
Since 75 = 7 * 10 + 5:
log7 75 = log7 (7 * 10 + 5)
Using the property log a^b = b * log a:
log7 75 = log7 (7 * 10 + 5) = log7 7 + log7 10 + log7 5
We can simplify further:
log7 75 = 1 + log7 10 + log7 5
Now, to find log7 10 and log7 5:
We can express 10 and 5 as powers of 7:
10 = 7 * (10/7) = 7 * (1.429)
5 ≈ 7 * (5/7) = 7 * (0.714)
Using the given logarithmic values:
log7 10 ≈ log7 (7 * 1.429) ≈ 1 * log7 7 + log7 1.429 ≈ 1 + log7 1.429
log7 5 ≈ log7 (7 * 0.714) ≈ 1 * log7 7 + log7 0.714 ≈ 1 + log7 0.714
Now, substitute the values:
log7 75 = 1 + log7 10 + log7 5 ≈ 1 + (1 + log7 1.429) + (1 + log7 0.714)
Approximating the values, we get:
log7 75 ≈ 1 + (1 + 0.429) + (1 + 0.245) ≈ 1 + 1.429 + 1.245
log7 75 ≈ 3.674
(c) To find log3 125:
We can express 125 as a combination of 3 and other numbers.
Since 125 = 3 * (3^2) * 3 = 3 * (3^3) = 3^4
log3 125 = log3 (3^4)
Using the property log a^b = b * log a:
log3 125 = log3 (3^4) = 4 * log3 3
Since log3 3 = 1:
log3 125 = 4 * log3 3 = 4 * 1 = 4
So, log3 125 = 4.
To evaluate the logarithms without using a calculator, we can use the properties of logarithms. Here are the steps to find the answers:
(a) To find log5 1.4:
First, we can use the property of logarithms that states log_a (mn) = log_a(m) + log_a(n). So we can rewrite log5 1.4 as log5 (14/10).
Next, we can use the property that states log_a (b/c) = log_a(b) - log_a(c). Applying this property, we rewrite log5 (14/10) as log5 14 - log5 10.
Since we know log5 7 = 1.209, we can write log5 14 as log5 (7*2).
Using the property log_a (mn) = log_a(m) + log_a(n) again, we rewrite log5 (7*2) as log5 7 + log5 2.
Since we also know log5 2 = log5 (10/5), we can substitute this value into log5 7 + log5 2, resulting in log5 7 + log5 (10/5).
Using the property log_a (mn) = log_a(m) + log_a(n) one more time, we rewrite log5 7 + log5 (10/5) as log5 7 + (log5 10 - log5 5).
Now, we can substitute the known values log5 7 = 1.209 and log5 5 = 1, resulting in 1.209 + (log5 10 - 1).
Since we don't know the exact value of log5 10, we cannot simplify it further without a calculator. So, the final answer for log5 1.4 is 1.209 + (log5 10 - 1).
(b) To find log7 75:
We can use the property of logarithms log_a (mn) = log_a(m) + log_a(n). So we can rewrite log7 75 as log7 (25 * 3).
Since we know log7 5 = 0.682 and log7 3 is unknown, we cannot simplify it any further without a calculator. So, the final answer for log7 75 is log7 25 + log7 3.
(c) To find log3 125:
Since log3 125 = log3 (5^3), we can simplify it using the property log_a (x^y) = y * log_a(x). Therefore, log3 125 = 3 * log3 5.
Since we know log5 3 = 0.683 and log5 5 = 1, we can solve this substitution as log3 125 = 3 * (log5 5 / log5 3).
Plugging in the known values, we have log3 125 = 3 * (1 / 0.683).
Evaluating this expression, we find that log3 125 ≈ 4.383.