Condense the expressions
1) 10 log x + 2 log 10
2) 1og base 5 of 8 - log base 5 of 12
Can you please explain to me how do you condense these expressions? Thanks in advanced!
10logx = log(x^10)
2log10 = log 10^2
10logx + 2log10 = log 100x^10
log8 - log 12 = log (8/12) = log(2/3)
To condense the given expressions, we will use the properties of logarithms. Here's how you can do it for each expression:
1) 10 log x + 2 log 10:
To condense this expression, we will use the property of logarithms that states log base b of a + log base b of c = log base b of (a * c). Applying this property, we have:
10 log x + 2 log 10 = log x^10 + log 10^2
Now, remember that log base b of b = 1 for any positive value of b. So, we can simplify the expression further:
log x^10 + log 10^2 = log x^10 + log 100
Using the property that log base b of a + log base b of c = log base b of (a * c), we can combine the logarithms:
log x^10 + log 100 = log (x^10 * 100)
Finally, we can simplify the expression by multiplying the terms:
log (x^10 * 100) = log (100x^10)
So, the condensed expression is log (100x^10).
2) log base 5 of 8 - log base 5 of 12:
To condense this expression, we will use the property of logarithms that states log base b of a - log base b of c = log base b of (a / c). Applying this property, we have:
log base 5 of 8 - log base 5 of 12 = log base 5 of (8 / 12)
Now, we can simplify the fraction within the logarithm by dividing:
log base 5 of (8 / 12) = log base 5 of (2 / 3)
So, the condensed expression is log base 5 of (2 / 3).
By following these steps, you can condense logarithmic expressions using the properties of logarithms.