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I am currently in high school and I'm in the process of preparing for my maths exams. One of the topics covered is algebra (exponents, logarithmic functions, binomial theorem, complex numbers). I am looking for a resource with problems of moderate difficulty. An example of what I mean by "moderate difficulty" is the following question related to exponents and logarithms:

$2 \log_{0.04}{(bx+28)} + \log_{5}{(12-4x-x^2)} = 0$ . Find $b$.

I am NOT looking for an answer to this question, it is just to provide some context. I am requesting additional resources as my math textbook has comparatively easier problems. If someone could share books/question-banks consisting of moderate or hard problems related to algebra, it would be great.

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Since you are not asking for an answer, I will give you an approach. First, note that there are two different logarithm bases. Notice that 0.04 = 5^-2. So logs to base 5 are -2 times logs to base 0.04. Once you have everything in the same base, you can take the second term to the other side of the equation cancel any common factors. You will have two logs that are the same, so their contents must be the same. (This is more advanced than high school math, but in college you will learn about monotonic functions, isomorphisms and other scary subjects.) Once you dispose of the logarithms, you will have an equation that you can solve by ordinary high school algebra to get a formula that allows you to calculate b if you know x.

I think your teacher was trying to push you to a higher level of abstraction: using the relationship between logarithms and their bases as a tool to simplify a problem. The more problems you attack, the stronger you will become. . . even if you don't always get the answer.

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    $\begingroup$ My impression is that the OP knows how to solve this and similar equations, and instead wants references to problems of similar difficulty to practice on. Also, the part about "This is more advanced than high school math" is fairly standard for high school, and I suspect pretty much every precalculus text states that $x = y$ can be concluded from $\log_b x = \log_b y$ (subject to suitable restrictions on $x,$ $y,$ $b).$ Indeed, this is immediate from the fact that graphs of logarithms satisfy the horizontal line test. $\endgroup$ – Dave L. Renfro Feb 11 at 18:19

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