How can the properties of logarithms be used to simplify and solve equations?

[schooler]

Use the properties of Logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms:
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[Ackbach]

What have you tried? Where are you stuck?

 

[schooler]

What have you tried? Where are you stuck?

For the first one I did " lnx-1/2ln(x^2+1)"

For the second one, I have no idea what to do :(

 

[MarkFL]

What should you do with the denominator? How about the product in the numerator?

 

[CellCoree]

The properties of logarithms are important in simplifying and solving equations involving logarithmic functions. One of the most useful properties is the product property, which states that the logarithm of a product is equal to the sum of the logarithms of each individual factor. This can be written as log(ab) = log(a) + log(b).

Using this property, we can rewrite the expression log(2x^3) as a sum of two logarithms: log(2) + log(x^3). We can then use another property, the power property, which states that the logarithm of a power is equal to the power multiplied by the logarithm of the base. This allows us to further simplify the expression to log(2) + 3log(x).

Additionally, we can also use the quotient property, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. This would allow us to rewrite the expression as log(2) - log(x^-3).

Finally, we can also use the fact that the logarithm of a constant is equal to a constant multiple of the logarithm of the base. This would allow us to write the expression as 3log(2) + log(x).

In summary, the expression log(2x^3) can be written as a sum, difference, and constant multiple of logarithms as follows: log(2x^3) = log(2) + 3log(x) = log(2) - log(x^-3) = 3log(2) + log(x). These properties of logarithms are crucial in simplifying and solving equations involving logarithmic functions.