solve the following logarithmic equation.
1. 2log(lower2) (x-2) + log(lower2)^2=3
I do not know how to make the 2 by the g on my computer.
2. log(lower729^x) + log(lower27^x) + log(lower3^x) =9
again I do not know how to make the numbers lower case
please show work, and if you can tell me how to make the lower case numbers on here.
lots of folks just write log_2(x-2)
the numbers are subscripts, not lower-case.
1. since we're dealing with logs base 2, just say log:
2log(x-2) + log2 = 3
log(x-2)^2 + log2 = 3
log(2(x-2)^2) = 3
2(x-2)^2 = 8
(x-2)^2 = 4
x = 0 or 4
I assumed that log(lower2)^2 meant log(lower2)2
2.
since 729 = 3^5, log base 729 = 1/5 log base 3
since 27 = 3^3, log base 27 = 1/3 log base 3
so, using all logs base 3,
1/6 logx + 1/3 logx + logx = 9
3/2 logx = 9
logx = 6
x = 3^6 = 729
To solve the first logarithmic equation, we will use the properties of logarithms and some algebraic manipulations. Let's go step by step:
1. 2log(sub2)(x-2) + log(sub2)^2 = 3
First, use the power rule of logarithms to simplify the logarithmic terms. The power rule states that log(suba)^b = b * log(suba)(c). Applying this rule, we have:
2log(sub2)(x-2) + log(sub2)(2^2) = 3
Simplify the logarithmic term on the right side:
2log(sub2)(x-2) + log(sub2)(4) = 3
Next, simplify the logarithmic terms further using the product rule of logarithms. The product rule states that log(suba)(b) + log(suba)(c) = log(suba)(b * c). Applying this rule, we have:
log(sub2)((x-2)^2 * 4) = 3
Now, we can rewrite the equation as an exponential equation:
2^3 = (x-2)^2 * 4
Simplify the left side:
8 = (x-2)^2 * 4
Divide both sides by 4:
2 = (x-2)^2
Now, take the square root of both sides (±√2) to eliminate the square term:
±√2 = x-2
Finally, add 2 to both sides to isolate x:
x = 2 ±√2
Therefore, the solution to the first logarithmic equation is x = 2 ±√2.
Regarding your question about entering the lower case numbers on the keyboard, it depends on the device you are using. On most standard keyboards, you can use the "Subscript" feature to enter lower case numbers. However, if you are unable to access this feature or use a device that doesn't support it, you can simply write the notation in a descriptive form (e.g., log base 2).