Given that log 2.5 = 0.3979, evaluate log 25 + 2log 250
To solve this expression, we will use the properties of logarithms.
Step 1: Recall the property log a + log b = log (a * b). Applying this property, we can rewrite the expression as follows:
log 25 + 2log 250 = log (25 * 250^2)
Step 2: Simplify the expression inside the logarithm:
log (25 * 250^2) = log (25 * 62500)
Step 3: Evaluate the product:
25 * 62500 = 1,562,500
Step 4: Rewrite the expression using the previously given logarithm value:
log 2.5 = 0.3979
Step 5: Evaluate the logarithm of 1,562,500 using the property log a = b:
log 1,562,500 = log (2.5 * 627,250)
Step 6: Apply the logarithmic property:
log (2.5 * 627,250) = log 2.5 + log 627,250
Step 7: Substitute the given logarithmic value and evaluate:
log 2.5 + log 627,250 = 0.3979 + log 627,250
Step 8: Finally, use a calculator to evaluate log 627,250 and add it to the previous result:
log 627,250 ≈ 5.7970
0.3979 + 5.7970 ≈ 6.1949
Therefore, log 25 + 2log 250 ≈ 6.1949.
Well, it seems like we've got some logarithmic fun going on! Let's break it down step-by-step.
First, we have log 25. Now we know that 25 is equivalent to 5 squared, so log 25 becomes log (5^2). Using the property of logarithms, we can rewrite it as 2 log 5.
Next, we have 2log 250. In a similar fashion, we can rewrite it as 2 log (5^3 × 2). Using the power rule of logarithms, we can further simplify it to 2(log 5^3 + log 2).
Now, to sum it all up, we have log 25 + 2log 250 = 2 log 5 + 2(log 5^3 + log 2). Can you guess what comes next? It's simplification time!
Using the rule of logarithms again, we can rewrite it as 2 log 5 + 2(3log 5 + log 2). This simplifies further to 2 log 5 + 6 log 5 + 2 log 2.
Finally, we combine the like terms to get 8 log 5 + 2 log 2.
And there you have it! The evaluation of log 25 + 2 log 250 results in 8 log 5 + 2 log 2.
To evaluate the expression log 25 + 2log 250, we'll use the properties of logarithms.
First, let's simplify the expression:
log 25 + 2log 250
Using the power rule of logarithms, we can write 25 as 5^2:
log 5^2 + 2log 250
Using the power rule again, we can rewrite 250 as 25 * 10:
log 5^2 + 2log (25 * 10)
Next, we can split the logarithm of a product into the sum of logarithms:
2log 5^2 + 2log 25 + 2log 10
Now, we can use the exponent rule of logarithms to simplify the expression further:
log 5^4 + log 25^2 + log 10^2
Applying these exponent rules, we get:
log (5^4 * 25^2 * 10^2)
To evaluate the expression, we need to multiply the terms inside the logarithm:
log (625 * 625 * 100)
Next, we can simplify the inside of the logarithm:
log (39062500)
Finally, we use the property of logarithm to rewrite the expression in terms of the given log value:
log 2.5 in the base 10 is equal to 0.3979. So, we can write 2.5 as 10^(0.3979):
log (10^(0.3979))^4
Therefore, the expression simplifies to:
4 * 0.3979
Evaluating this expression gives us:
1.5916, which is the final answer.
log25 = log(10*2.5) = log10 + log2.5 = 1.3979
log250 = 2.3979