10^(-10log3)
How to solve this???
one of the log properties is .
a^(loga k )
= k
we have that case, if we change
10^(-10log3)
to
10^(log 3^-10)
we get 3^-10 = 1/59049 ---> exact answer
or .000016935
( if I use my calculator
10^(-4.771212547)
= .000016935
do u know how I could find this in a way that creates a fractional form?
As Reiny noted,
3^-10 = 1/59049
To solve the expression \(10^{-10\log_{10}3}\), we can break it down step by step using the properties of exponential and logarithmic functions.
1. First, simplify inside the logarithm. Since we have \(\log_{10}3\), this represents the exponent to which we raise 10 to obtain 3. In other words, \(\log_{10}3\) is asking "what power do I need to raise 10 to in order to get 3?". So, \(\log_{10}3\) is equal to the value \(x\) that satisfies \(10^x = 3\).
2. Solving \(10^x = 3\) is not straightforward, but we can approximate the value of \(x\) using a calculator or mathematical software. On most calculators, you can use the "log" button (without any subscript) to calculate the logarithm to base 10.
By using a calculator, you can find that \(\log_{10}3 \approx 0.477\). This value can be slightly different depending on the level of precision of the calculator.
3. Substitute the approximate value of \(\log_{10}3\) back into the original expression: \(10^{-10\cdot 0.477}\).
4. Compute the value of \(10^{-10\cdot 0.477}\) by raising 10 to the power of \(-10\cdot 0.477\). On most calculators, you can use parentheses to ensure the correct order of operations.
By evaluating \(10^{-10\cdot 0.477}\), you will find that it is approximately equal to 0.00000002. Again, the exact value may differ slightly depending on the calculator used.
Therefore, the value of the expression \(10^{-10\log_{10}3}\) is approximately 0.00000002.