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To solve the problem , we need to express the number 8 as a power of a base that simplifies the logarithm. We can write 8 as , because 8 equals 2 multiplied by itself three times.
Let's use the power property of logarithms, which is:
Applying this property to , we have:
Using the power property, this becomes:
Therefore, the expression for in terms of is:
.
\( \log_{10}3+\log_{10}4= \)
While calculators give decimal approximations, the question asks for an exact algebraic expression. The answer is more precise and shows the mathematical relationship.
Look for perfect powers of small integers! Since 8 = 2×2×2, we write it as . This makes the power rule applicable and simplifies the expression.
Don't worry about connecting base 6 and argument 8 directly. The key is expressing 8 as a power of any convenient number, then using logarithm properties to simplify.
The power rule is the most useful here since 8 is a perfect cube. Other properties like product or quotient rules would make this problem more complicated.
Remember that asks "what power of 6 gives 8?" Since , our answer works!
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